In Inquisitor, a spin off of Warhammer 40,000, you can choose to aim for an opponent's groin. It causes a fair bit of damage, and leaves them almost useless for a turn. It's funny.
Some players have remarked about unusually accurate shots from meltaguns ("weaponised microwave oven" is a fairly accurate description) and chainswords.
In the Ork Codex, a mortally-wounded Warboss did this to a Daemon Prince as a kind of "parting shot". With a Power Klaw. Khorne was so impressed he resurrects the orks every day on that planet so they can fight and die and fight again. It's win-win.
Not quite played for drama or comedy, but Gabriel Seth of the Flesh Tearers Space Marines has a special rule that means he gains one attack every time an enemy unit rolls a 1 on a D6 when rolling To-Hit. This is supposed to represent him doing this along with headbutting and throat-biting. Yep. He's just like that.
I.C.E.'s Cyber World had the following entry in the "Critical Failure: Melee" table: "Worst move seen in years. You take 60 pts of damage to your own groin. Opponent stunned for 3 rounds with laughter."
In Warhammer Fantasy Roleplay, the Bretonian truffle hound is noted as becoming extremely jealous of any males in the vicinity after eating a truffle, and will take steps to rectify the situation. The consequences are not spelled out, but are still pretty obvious.
In GURPS, this is very effective, causing twice the normal shock penalty.
If you're small enough, the game allows you to upper-cut an enemy in the groin.
There's also the "Testicle Grab" technique.
If you randomly attack someone the groin is the second most likely part of the body to be hit.
Werewolf: The Apocalypse has one entry on the Battle Scars table labeled simply "Ahem..." The injured spot isn't named, but it's specified that you are now immune to seduction attempts, unable to seduce others, and (if male) sterile.
In Hackmaster, gnome titans have a special "groin stomp" attack, and characters can perform a "groin punch" attack.
The Celestial Monkey martial arts style in Exalted lets you make a called shot to an opponent's primary virtues, as represented by areas of the body. In Valor's case, said area is the groin (something of a pun on "balls of steel"). This does up to five levels of lethal damage depending on the enemy's courage, which is impressive considering that just-starting Exalts and heroic mortals only have seven. (You can punch a moderately-brave extra in the crotch so hard he dies. Never let it be said that Exalted does anything by halves.) This is bonus damage in addition to whatever damage got past their soak, making it possible to kill an Exalt with a kick to the nuts.
Perhaps showing The Monster Squad as one of its influences, Grimm has an advantage that can be taken by PCs who already have the advantage "Dirty Fighting" called "Right In The Nards!", and the rules on combat feature an illustration of a kid kicking a wolf-man in the crotch.
The fan created netbook "Guide To Unlawful Carnal Knowledge" for AD&D features "The Rack Critical Hit Table," which shows the various results of a male character getting "racked."
Paranoia supplement Acute Paranoia, adventure "Miami Laser". At the end of the adventure, after the "Miami Laser" show replaces the Teela O'Malley show, any surviving Troubleshooters are attacked by rabid Teela O'Malley fans. One of the attacks they use is a knee to the groin.
Implied by the "Psycho Squirrel" monster in Munchkin, as it will not attack females, or characters wearing the "spiked codpiece" item. And its "bad stuff" is "speak in a high squeaky voice until next turn".
Played for Drama
Seth Farrow, recurring villain in Scion, took a shotgun blast to the groin the night he murdered his nephew Horace's family. It's never stated outright, but it's clearly implied he lost his genitals; most notably, when a captured Donnie Rhodes greets him with "How they hangin'?", Seth pulls a knife on him (Kane barely keeps him from using it).
Iron Crown Enterprises: Any location based damage in HARP or Rolemaster with a hit location of "Abdomen and Groin".
Shadowrun. In the short story "Plus ça Change" at the beginning of the 2nd Edition main rules, the street samurai Nameless kicks a Rat shaman in the groin to prevent him from casting a spell.
Dungeons & Dragons, Dungeon magazine adventure "Flowfire". A half-ogre named Thrash Groin-Guard has a spike on the knee of his armor. He can strike twice with the spike per round, aiming for his opponent's groin. To add insult to injury the spike is poisoned, doing 1-6 additional Hit Points of damage per minute.
According to the Vampire: The Masquerade book House of Tremere, the Tremere stronghold of Ceoris was actually created through one of these. Back in the days when Tremere and his followers were still mortal mages, the most powerful of his House gathered in the mountains of Transylvania to ritually consecrate the site of their intended chantry: said ritual commenced with Goratrix willingly castrating himself with a sickle and using his own severed genitals as a sacrificial offering. For good measure, after Ponticulus had to be sacrificially decapitated for accidentally interrupting the ritual, Etrius decided to compound the sacrifice by putting Goratrix's amputated penis in the severed head's mouth note with the intention of binding the ghost of Ponticulus to the site. And in a final insult, Goratrix - who'd only committed to this act of sacrifice in the knowledge that magic could heal these kind of injuries - discovered that his regeneration left a lot to be desired.