If you're looking for PSP games for kids, you'll quickly discover that the PSP hasn't really been marketed to kids -- the main target audience seems to be teenaged and adult males -- but there are kid-friendly games to be had if you look closely enough. Here are some games that kids will find fun and parents won't be dismayed by.
Ben 10 is an animated TV series for kids, about a boy with alien powers who uses them to help his friends and save the Earth. There are actually 3 PSP games based on the show: Ben 10: Protector of Earth, Ben 10: Alien Force and Ben 10 Alien Force: Vilgax Attacks. All 3 are rated E10+ (Everyone 10+) by the ESRB, due to cartoon violence, so you might not want to give them to your younger kids. Ben 10 is more of a boys' show than a girls' show, but a lot of girls like good action-adenture games, too. You can see screenshots from this game in the gallery.
One of the most beloved (and kid-friendly) series on the full-size PlayStation is the Jak and Daxter series. The character Daxter is a small, furry, orange "ottsel" (sort of like an otter, sort of like a weasel) who is the main character's sidekick, but in the PSP game Daxter he takes the leading role. The game has the player evading guards and killing dargerous insects in order to (eventually) rescue Jak from the evil overlord's clutches. There is some shooting (of insects, mostly), so it's not a game for families opposed to guns, but for everyone else over the age of 10 (and I definitely include adults) Daxter is great fun. You can read my review for more information.
Though the little girl protagonist and the girly graphics may put off boys, this game is really great for all ages and both sexes (I even recommend it for adults). It's rated E10+ in North America, due to cartoon violence, but the message in this game is really one of friendship, so your under 10 kids might be just fine with it. Gurumin has a simple control scheme and a fairly even difficulty curve, so it's great for newer gamers, but it's just so very much fun that I think everyone should play it. For more information, read my summary review and my full review, and for screenshots see the gallery.
LEGO is one of the most popular toys around for both boys and girls. Combine LEGO with just about anything else and make it a videogame, and you've got a hit. All of the LEGO videogames are pretty good, but Indiana Jones is my personal favorite. There is a certain amount of violence (of the cartoon sort), earning the game an E10+ rating from the ESRB, but everything--including enemies--is made of LEGO and turns into a pile of bricks when hit, so it's hard to object. You can read more about this game in my review, and look at screenshots in the gallery.
With a rating of E for Everyone, you know Manic Monkey Mayhem is pretty safe for just about any kid. It's a very silly game, about monkeys flinging bananas at each other, but it's simple, easy to pick up, and fun. And, because it's a PS Minis game, it's very inexpensive. You can't buy it in a game shop, though, as it's download-only, so you'll have to figure out how to navigate the PlayStation Store (luckily, Sony makes it pretty easy, with a link to the store right on the PSP's main menu). Find out more about this game in my review.
Pipe Mania is a frantic-paced puzzle game that has the player laying out plumbing to get liquid (or rubber ducks!) from start to finish. Very young kids will probably find the later levels too difficult, but then again, they may surpise you. This is a great game for teaching kids how to plan ahead, and there's really no violence at all. Plus, it's fun. Read more about Pipe Mania in my review, or have a look at screenshots in the gallery.