Wendy Garret of Gun X Sword is a teenage example of this. She very rarely helps out in battle, but she's better at thinking than Van is. She functions as a manager, handling the money and making travel arrangements.
Eve of Black Cat is one, but she hardly has trouble keeping up.
Robin, sidekick of Batman, is the Ur Example, and is retained in almost every incarnation of Batman, no matter how Darker and Edgier, simply because of tradition. These days the concept is subject to Reimagining The Artifact, justifying Robin's presence by saying that either the kid is so damaged that he'd become self-destructive (or just plain destructive) without Batman supervising his crimefighting (probably the case with Jason and definitely for Damian), or that Batman is the damaged one and needs a surrogate son (all but said was the case for Tim).
In The Dark Knight Returns, when the new police commissioner tangles with Batman and spots the young Carrie Kelly as Robin with him, she adds Child Endangerment on the warrant for him.
Following the Batman formula, other younger sidekicks to adult superheroes include Supergirl, Speedy to Green Arrow, and Wolverine's various sidekicks, usually of the young girl type. (Kitty Pryde, Rogue, X-23, Jubilee)
Also on the sidekick squad: Aqualad to Aquaman. Kid Flash to Flash.
DC thrives under this trope. Darkly parodied with Kid Devil, whose adult counterpart and idol Blue Devil was barely aware of his existence, despite having made a Deal with the Devil to be a superhero with him.
The Clock, a very first Golden Age hero (last seen in 1944) and his kid sidekick Butch (a girl). Butch, incidentally, originally wanted to be the Clock's "moll", much to the hero's mortification. She eventually got over it.
Johnny Bates, AKA Kid Miracleman, from the comic Miracleman. (Originally known in the UK as Kid Marvelman and Marvelman.) Alan Moore's 1980s revival of the series not only aged the character to adulthood but gave him one of the most horrific Face Heel Turns in comic book history. There was also the Teen Sidekick Young Marvelman.
Spider-Man started as a deliberate subversion of this trope. Stan Lee had grown sick of teen sidekicks, so he decided to create a series that featured a teenager as the main star, rather than as support to an adult hero. Everyone thought that It Will Never Catch On...but it did.
Last Action Hero even has the Kid Sidekick refer to himself as the 'Comedy Sidekick', which should be a sub-trope of sidekick.
The Jedi in Star Wars pair up young padawans with older Jedi to teach them the way of the force. The difference here is that they often remain padawans into their twenties, making them former Kid Sidekicks by that age. They are usually Knighted after that, and soon begin the cycle again by choosing their own padawan. Generally padawans and their masters are sent on diplomatic missions, but with the way the Star Wars universe works, and with the advent of the Clone Wars, it's almost a given that the padawan will be forced to fight.
In Wearing The Cape, Hope (18 years old) becomes Atlas' sidekick in order to learn the butt-kicking ways of Atlas-type heroes. The whole mentor/sidekick angle is played up for the media (her costume is even color-coordinated to match his), but it's clearly understood to be a temporary arrangement, more like an apprenticeship.
Live Action TV
Mark Hollander to the titular Ace Lightning (albeit sometimes reluctantly).
Rather bizarrely on Heroes, Sylar temporarily gained one of these in the form of Luke, a troubled teen with the inexplicable desire to go on a road trip with a super-powered serial killer. And keep annoying him. His survival is something of a miracle. The series strongly implied at the time that Luke is Sylar's brother. As with most things on the show, this turns out to not have been the case when it's later learned that Sylar is a Petrelli.
In Nickelodeon's U-Pick Live segment, they held a contest where the winner would be added to the show as Pick Boy's new sidekick. Something must have gone wrong backstage though, because no kid was ever brought in.
On Doctor Who, Adric filled this role. Susan and Vickie were likewise younger and less mature than most. Of course, given the Doctor's full age, almost every companion is a kid to him, unless they're Timelords too.
Maya Fey, the main assistant in the Phoenix WrightAce Attorney games may or may not count, as she starts 17 but becomes 19 by Trials and Tribulations, although she tends to act like a little kid sometimes. Phoenix also had Ema Skye in one case when she was 16, and Apollo Justice has Trucy, who is 15.
8-years-old Pearl Fey may also count in the missions where she plays a sidekick to Phoenix.
It's tradition for the protagonist of the Ace Attorney games to have a cute young girl for a sidekick. Or Gumshoe. Even Edgeworth teams up with Kay Faraday... and then it gets sent up when he's helped by FranziskaVonKarma. (who also briefly and hilariously becomes Phoenix's sidekick in the third game)
Miles 'Tails' Prower, sidekick of Sonic the Hedgehog, is of the Gadgeteer Genius variety and has abilities that complement Sonic's. Later games introduced Cream the Rabbit (who has her own Sidekick, the Chao named Cheese), who serves as more or less the female counterpart to Tails (hence why they are often shipped together, despite almost never interacting). She started as a sidekick to Amy, but in Sonic Rush she acted in this capacity to Blaze. In Sonic Rush Adventure, Blaze gets her own sidekick in Marine the Raccoon.
The Silver Age-inspired Freedom Force computer games give us Liberty Lad, sidekick to Super PatriotThe Minuteman. Young Nick Craft, founder and president of the Freedom Force Fan Club, tags along after the team as they try to bring down mobster-turned-supervillain Pinstripe. Naturally, he gets shot.
Pinstripe: Now youse gotsta choose, heroes. Come afta me, or save da brat! Mentor reports that he needs an immediate transfusion... or he'll die! Minuteman, feeling responsible for the plucky youngster, valiantly volunteers to give his own blood to the boy. Mentor warns him that they have no idea what will happen, because Minuteman's blood is infused with... Energy X.
Minuteman: By the Constitution, Mentor, there's no time!' The Energy X in Minuteman's blood gives Liberty Lad super strength and agililty and a penchant for red, white and blue tights. He's one of the more fun and effective characters in the game.
And from the same game, we have Man-O-War and his preteen sidekick, The Sea Urchin.
Gordito Delgado from The Adventures of Dr. McNinja. The Doctor even compares himself to Batman when he decides to take Gordito in. Of course, the Doctor compares himself to Batman regardless. Note that Gordito's origins - a child of a family of circus performers who was orphaned mid-show - is a tribute to Dick Grayson's.
Nip and Tuck has an arc that shows a Serenity-inspired movie featuring a kid sidekick.
In the backstory of the Whateley Universe, the pre-World War II superhero Champion had two: Miss Champion and Champion Junior. In a Shout Out to the way heroes age in Comic Books, Miss Champion only aged (from then on) at about one-fourth normal, which really messed up her social life. Miss Champion is still around, and one of the important side characters of the universe.
Spoofed with Barnacle Boy, Mermaid Man's sidekick on Sponge Bob Square Pants, who is still treated as a child even though both of them are now old men. One episode has him doing a Face Heel Turn because of it.
The Transformers have a bad habit of dragging young humans into their battles - young humans who tend to have no powers or skills that make letting them within a mile of Decepticons anything less than criminal negligence. Giving them a reason to be around at all is relatively new to the franchise.
Young Justice has the main characters being sidekicks to members of the Justice League (Robin to Batman, Aqualad to Aquaman, Kid Flash to The Flash and Speedy/Red Arrow to Green Arrow). Other characters who aren't sidekicks in the comics canon are reimagined as such here, such as Miss Martian being Martian Manhunter's niece and Bumblebee being the Atom's student, while Beast Boy is linked to Miss Martian. The only character who doesn't have a mentor is Blue Beetle... because the previous Blue Beetle was killed.
The Cadets in Voltron Force are a fairly good example. They bring considerable talents to the team - Vince's Technopathy, Larmina's martial arts talent and Daniel's piloting skills and agility (both physical and mental) complement, rather than overshadowing, the rest of the Voltron Force.
Sidekick the 4 main characters are kids in sidekick school, who will soon be sidekicks to superheroes, Eric hopes to be the sidekick of his superhero Maxum Man.
Me and the others gave everything, but it just wasn't enough for the old man. I used to think, if I went on long enough, someday he'd retire and I'd... ah, the heck with it. Capes, costumes, bad guys - it was kid's stuff! Bruce probably did me a favor. By the end, I was so sick of it I never wanted to see that stupid Robin suit again...!