In Dragon Ball Z, Chi-Chi and Gohan have an argument when Gohan wants to go to Namek. Chi-Chi made a good point when she said that Gohan should be having fun as a kid instead of trying to be an adult, and Gohan also had a point since his priority was Serious Business.
In Karakuridouji Ultimo, the protagonist Yamato learns he is the cause of an apocalypse in the near future. (This is part of the premise, so it's not really a spoiler.) Yamato chooses to avert this by finding every person in the world who would be involved in the event and understanding their points of view so that he can choose the best possible action once the time comes.
A lot of Hayao Miyazaki's films are based on this kind of premise. He dislikes the limiting assumptions of a lot of conventional media that evil exists and must be defeated by good.
Princess Mononoke is perhaps the best example of this, with every character having a reasonable explanation and motivation for their actions. San is harsh and violent - but only wants to protect her home, family and the natural world. Eboshi wants to kill the god of the forest - but is a benevolent leader, good to her people and kind to lepers and you can fully see why her people are willing to die for her. The protagonist Ashitaka is completely neutral and genuinely wants the best for everyone. Even if the consequences of their choices are ultimately negative, you can see why they did it.
The Social Network is done this way, and the characters themselves reach this conclusion: None of them is truly unsympathetic, and they all have more or less valid claims and complaints.
In Team America: World Police, both 'dicks' and 'pussies' have a point, according to Gary's (plagiarized) speech at the end. The 'assholes' on the other hand, just make everything worse for everyone.
X-Men had this trope for the political hearing which Jean Grey debated with politicians concerning mutants. Both sides brought up good points which was the intentions of the director.
The Brotherhood vs. X-Men conflict in general is this, especially in The Last Stand; when a cure is introduced, Magneto is primarily wary that humans will 'draw first blood' and use it to forcibly strip mutants of their powers—which is exactly what they do. Unfortunately, he, the Brotherhood and Jean Grey then go on to launch an attack on the cure-production facility (tearing the Golden Gate Bridge off its foundations in the process) with the stated intent of destroying the cure's source — which happens to be an innocent teenage boy who is himself a mutant, thus giving the humans every reason to believe the mutants are exactly as dangerous and destructive as feared.
In Traffic there are more than just two sides and most of them have a point. The most surprising one coming from the arrested drug dealer who points out that the DEA agents are also technically working for the drug mafia as they are being used by one of the drug cartels to destroy its opposition.
What makes WarGames so special is how it delivers its anti-war message without demonizing either side. Both General Beringer and Professor Falken have sound ideas about how best to deal with JOSHUA and prevent him from causing a pre-emptive nuclear war.
The trope is reduced to absurdity in an old Jewish joke. Two Jews come to a rabbi to resolve a dispute and present their arguments; they also bring along a witness. The rabbi, after leafing through the Talmud for a couple of hours, finally says: "Shlomo, you are right. But, Moyshe, you are right as well". The puzzled witness asks: "But, rabbi, how can two men with completely different opinions be right at the same time? It's impossible!". The rabbi replies: "You know, Joshua, it turns that you are right as well!"
And the spinoff joke, in which two supplementary angles go to the rabbi because neither one wants to admit to being obtuse. The first angle presents its case, and the rabbi says "It seems you are right." The second angle then presents its case, and the rabbi says, "You're right, too."
The Sheriff of Nottingham is able to do this to himself in In A Dark Wood, Michael Cadnum's Good Versus Good retelling of Robin Hood. Halfway through the book, he is able to recognize that although Robin Hood is an outlaw, he is also a good man. It isn't until the end of the book that he is able to find a point of reconciliation between this and his duty to uphold the law.
The Audacity of Hope by Barack Obama use this trope as a cornerstone for much of it's portrayal of the political landscape.
Black Crown: In the story 'Schism', both King Flavius and Lord Corrigan have a point depending on how you view a government's duty to its people.
Frasier: From the episode Dinner At Eight; one the one hand, Martin is right that Frasier and Niles need to relax and enjoy "normal" things sometimes; but seeing as how their hostess just cut off their (presumably expensive) ties, they have a right to be angry. Or at least very, very annoyed.
The later versions of Mage: The Ascension used this perspective. The Technocratic Union wants a stable and democratic reality where everyone is able to create miracles through technology. They have largely succeeded: the modern world with computers, airplanes and modern medicine exists by their design. Their opponents, the Traditions, prefer a more unstable (ahem, dynamic) reality with more personal freedom - a freedom of expression that includes rewriting reality itself rather then merely writing words. (The original version had this same conflict of interest, but hardcoded that the Technocracy's ideals made them Dirty Communists.) Taken a step further in The Sorcerer's Crusade; in the Dark Ages, when everyone believed in magic, the Traditions (especially the Order of Hermes) were in control and maintaining the stability of the world while the upstart Order of Reason wanted to overturn this consensus reality for the good of everyone. The future Technocrats won, at which point the Traditions became the Plucky Underdogs.
The successor game, Mage: The Awakening, took this conflict and put it into the tradition mages and the free council, both playable factions that are nominally allied and can be mixed into the same cabal. The traditions essentially advocate the "magic is superior/new things are bad" viewpoint, and are backed up mechanically by new human technologies draining the power from old spells. The free council thinks new stuff is great, and even works on drawing power from new concepts and even technology in crafting rote spells.
To clarify, the traditions are correct in that mundane humanity is draining the old magic from the world, and the free council is correct in that humanity is producing new wonders which have value in themselves.
The primary antagonist faction was changed in Mage: The Awakening into decadent god-priests obsessed with controlling the destiny of mages and mortals specifically to avoid this trope, though many complained that it weakened the setting by bringing up the trope in a new way: that's pretty much what the traditions want, too.
Pretty much a running theme in White Wolf games: every splat has an antagonist faction where the conflict is black v grey, e.g. you're a werewolf fighting the wyrm to keep the world from being destroyed, a faction where the conflict is grey v grey, e.g. you're a mage in conflict with the technocrats because you're competing for a prize you both want (control of reality), and a faction where the conflict is white v grey... and you're the grey, e.g. you're a vampire and human hunters are coming for you with stakes (and are completely in the right, as even if you're "moral" you're one failed will save from a murder spree and have probably failed that save at least once).
Dungeons & Dragons has settings that try to take this approach to racial conflicts, as an alternative to Always Chaotic Evil and everyone just being tragically misunderstood (which gets bland quickly). Eberron, for instance, affiliates most of the "monster" races with fallen or current civilizations with religious or economic conflicts with the players' home civilizations. This dodges the unfortunate implications of the usual racial warfare while still giving reasonably simple reasons to get into sword and sorcery brawls within moments of running into each other.
Fiddler on the Roof runs on this trope. Tevye is caught in the clash between the traditional world and the modern world. He's a really smart guy, but poor and uneducated. He tries his best to be fair and see both sides of the situation, with many inner monologues about "on the one hand [...] but on the other hand". In the page quote above he gets ridiculed for not simply picking a side when two guys who both have valid ideas stick to parroting slogans at each other instead of making more nuanced arguments for their causes.
In The Amazon Trail 2, one location has you talk to an oil executive and a native from the area. The executive wants to drill for oil, and the native doesn't want the environment to be spoiled. Now while the game has a bias to the native, the game will only reward you if you listen to both characters about the issue.
This is the Paragon resolution of post-loyalty mission conflicts between members of The Squad in Mass Effect 2.
Inverted in the Renegade resolutions, where Shepard points out that neither side has a point and they're both endangering the mission for no good reason.
The Templar/Mage conflict in Dragon Age II is the epitome of this trope. The Mages are horribly oppressed by the Chantry's Templars, imprisoning them to keep the city safe and treating all Mages as dangers. At the same time there are a lot of Mages who seem to turn to Blood Magic and the like, due to the weakness of the Veil in the area. Better safe than sorry?
A major theme of the Dragon Age universe as a whole, really, there is never a clear-cut right or wrong answer to any conflict in either game.
This is what drives a lot of the Grey and Gray Morality in the Geneforge series, with even the more "evil" factions such as the Takers or Barzites making the occasional valid point.
The Stormcloaks are correct in that their traditional religion is being unjustly oppressed (more or less — they phrase it as being a tad more traditional for Nords than is actually the case, but it is still both centuries old and unjustly oppressed), that Ulfric is a man of great courage and loyalty to his allies (which his enemies acknowledge) and is a paragon of traditionalNord values, the Empire caved in in order to end the Great War (in which thousands of Nords gave their lives fighting for the Empire), and that it's become a decrepit, corrupt entity that has given their enemies huge amounts of power in Skyrim without the Nords' consent. Also, a few characters mention that the Empire has been putting high taxes on Skyrim after the war, limiting the citizens' well-being.
The Imperials are correct in that the Stormcloaks are full of xenophobic assholes, Ulfric used a traditional challenge as an excuse to effortlessly kill the previous king (albeit with the victim accepting his challenge,) and his attempt to seize power may well be that of a power-hungry tyrant, the Empire's capitulation was the only way to prevent even more death and destruction for every race and culture, and that if the Empire starts to come apart it will be easy pickings for their enemies, who both sides agree are evil, and openly have the intention of dominating and oppressing the world in any way they can. Also, some characters reveal that the unity of the Empire allows for prosperous trade and surplus of food, and Skyrim's independence will damage this trade.
There's something of a Deconstruction in KotOR II. Inside Ludo Kressh's tomb, the player is faced with a series of illusions. In one of these, the player's companions are about to attack Kreia, (the player's mentor) and the player must decide who to side with. However, if the player answers "I won't attack you, but I won't stop the others from attacking you either", Kreia exasperatedly scolds you and everyone present tells you that "apathy is death".
RuneScape: Word of God is that each of the setting's gods have their own idea of what is best for the world, each as valid as the next. Even Zamorak, previously considered the God of Evil.
Rift gives us the Guardians vs. the Defiants. The Guardians are divinely inspired/powered heroes seeking to rid the world of the Defiants whose technology is destroying the only thing keeping the dragons from causing The End of the World as We Know It. The Defiants are persecuted "lesser" races who have dealt with the Guardians bearing down on them for generations, and who can only defend themselves with said technology. Both sides are right, and both sides are at fault: the Guardian intro makes it clear that the Defiants are, actually, at fault for unleashing Regulos and the dragons; however they go way too far in their persecution and drive some members of the Defiants to make a Deal with the Devil in the first place, and neither side can stop killing the other one all the way through the far-future Defiant intro to realize they shouldn't be enemies, even as the world literally dies around them and Regulos gets his way. The character of either side ends up in the "middle ground" timeline of the main game (either because of being Not Quite Dead for years, or through actual Time Travel) at which point they immediately go about killing the other side instead of actually dealing with the Dragons for most of the early areas.
T-Rex: He says - there's some merit to both sides of the issue?
Done on The Simpsons in the episode "The PTA Disbands" when Principal Skinner and Mrs. Krabappel are trying to convince the parents at a PTA meeting of their respective positions. Mrs. Krabappel argues that Skinner's budget cuts are harming the education the parents' children are receiving, and that they need the resources to do their job. The parents are inclined to agree with her until Skinner points out that the school is on a very tight budget as it is, and for the school administration to get what the teachers are asking for they'd have to raise the parents' taxes. That gets the parents complaining about taxes being high enough as it is, and the debate between Skinner's and Krabappel's positions ends up going back and forth. The episode ends by Skinner and Krabappel deciding to Take a Third Option and rent out the school's cloakrooms to the prison system to raise extra money, although the writers don't provide an answer to the taxes vs. education quality debate.
Done occasionally on Futurama as well. Matt Groening was once quoted as saying that he believes satire works better when you give both sides a valid argument.
South Park uses the Golden Mean Fallacy a lot to find a middle ground between two opposing sides, ultimately arguing that each side is partially correct, making it all the funnier when this attitude is subverted, such as the NAMBLA episode which has the leader of NAMBLA making a long-winded speech about what is wrong and right followed by a short retort from Stan which undoes the speech entirely.
The animated TV adaptation of The Lorax does acknowledge that a lot of people would lose their jobs if the Thneed factory shut down.
Doug: After completing his last Quailman comic, Doug is caught doodling and making fun of Lamar Bone, who really had it coming. Doug knows Bone didn't get his word on his work, and he simply traps Bone by asking if he was immune to the rules he enforces. Bone explains that he wasn't, cue this trope and the deal Bone makes with the students adding a rule that bans Saturday detentions. Bone doesn't like to be made fun of, but his actions can also get him in trouble.
The duet in the My Little Pony: Friendship Is Magic episode "Bats!" demonstrates that while Applejack is rightfully worried about the state of the orchard if they don't do something about the vampire fruit bats, Fluttershy is also correct in that they provide long-term benefits if left alone.
The show does like to provide this a fair amount of times, arguably preferring to teach kids friendship is about compromise and respect on both ends to avoid previous unfriendly aesops. "Lesson Zero", "Over The Barrel" and "Sisterhooves Social" are prime cases both sides admit the other has valid arguments and end up taking a middle road.
This is the reason some people prefer to use the Golden Mean Fallacy when considering controversial topics like "Which political party really is worse for the country" or "Evolution vs. Creationism" and so on.
It's also why political power can and frequently does swing back and forth between different parties in democratic countries, as voters decide they like one party's policies at one time and then decide to switch to another party's policies later on. Sometimes parties who win elections and form governments end up plagiarizing parts of their opponents' platforms in order to broaden their own appeal in the electorate.
Similarly, in the Canadian 2011 election, the arguably most centered party (Liberal) was squashed out in favour of the NDP, which is, for the most part, more to the left, and the Conservative party, the right-most major party, which was already strong beforehand. It's up to debate how much of this was because of Jack Layton, and how much it was because of the poor showing of the Liberals (including the fact that they didn't claim Both Sides Have a Point).