The PSP and puzzle games are a perfect fit. By nature, puzzle games tend to be best played in short sessions, which works well with the PSP's portability. Consequently, there are quite a few puzzlers available for the PlayStation Portable. Here are some that I've played and enjoyed. I'll update this list as I find new games I think are worth mentioning, so check back from time to time.
Lumines was the first game I got for PSP, and it's still one of my favorites. Flashy backgrounds and groovy music get flashier and groovier the better you play, and higher scores unlock new images and new songs. It's a simple, Tetris-inspired game that is a snap to pick up and play (I sat down to just give it a quick try when I bought it, and looked up to realize and hour had gone by in an instant). Lumines II adds a few features for an even better game. Either of the two would be a good choice, or you can start with the first and then move on to the second.
This game is classified as a "puzzle-rpg," so I could just as easily have included it on a "Top RPGs" list. The puzzling forms the core of the gameplay, though, so it gets to be on this list. Puzzle Quest has been a hit on every system it's been on, and for very good reason (though I only have it for PSP, I've been tempted more than once to buy other versions). The main puzzle-play is sort of Bejeweled-inspired, but Puzzle Quest uses its RPG elements to add some really interesting and challenging aspects to the basic game. Spells, mounts, and the strengths of different characters each add something a little different, leading to a highly addictive game.
I used to spend far too much time playing demos of Zuma on my laptop, so when Luxor--like Zuma but Egyptian instead of Aztec--was released for PSP, I was delighted. And I haven't been disappointed yet. This is another one of those games that suck you in for longer than you meant to play ("Just one more level," you might think to yourself.). The Wrath of Set is the older of the two games. Pharaoh's Challenge adds a few new elements, but probably not enough that you need to buy both games. Unless you're a Luxor/Zuma fiend like me.
Move your cube from start to finish. It's a simple premise, but more obstacles--and more difficult obstacles--make each level more complicated than the next. I didn't find Cube as absorbing as some of the games on this list as it requires a lot from your brain, but it's perfect for short play sessions. Simple graphics and gameplay give it a bit of a stripped-down feel, but it suits.
While 7 Wonders of the Ancient World isn't a revolutionary game by any means, it is a solid entry in the match-3 genre, with a neat ancient-world theme. It even has historical notes on each of the seven wonders you "build" as you play the game, so if you want to educate yourself as well as play, you can.
This remake of a 1989 game shows that a simple premise can take a good game a long way. Pipe Mania has been remade before, for various platforms--this version ads updated graphics and new gameplay elements to an already addictive formula. I found it incredible frustrating at times but also very rewarding. A bonus is the inclusion of the original game (albeit in graphically-updated form).
More "get from A to B" games, Mercury and Mercury Meltdown make that simple goal more complicated through various kinds of obstacles and difficulty levels, and because the thing you have to move is a blob of liquid that has a tendency to come apart into separate drops that like to roll off in different directions. Of the two games, I'd recommend the second--Mercury Meltdown--only because the new elements improve the gameplay. If all you can find is Mercury, though, it's well worth a play, too.
PQ stands for "Practical Intelligence Quotient," and the game is a sort of IQ-test like thing. Don't let that put you off, though. Like some of the other games on this list, PQ and PQ2 challenge you to get from the starting point to the end. The puzzles are timed, and they get more and more difficult with each one you complete. Not only do the puzzles themselves get more complicated, but "police" of varying levels of intelligence start to appear (get caught in the beam of their flashlights and you lose the puzzle). Each puzzle gets you a score based on your time and the numbers of moves you need to get from start to finish. These scores are compiled as you go into an overall "PQ score" that you can compare with other gamers.
I haven't reviewed this one--I didn't really get far enough for a review before having to set it aside for more pressing titles--but I'm having a lot of fun with it when I do pick it up. The "find the exit" gameplay in Exit is enriched by the addition of obstacles, naturally, but also by other characters that you have to usher to said exit. Frequently you have to enlist the help of some characters to get others out. The stylish graphics add a lot of appeal to these games. I've only played the first, but Exit II looks equally as good.