Here's a few choice suggestions rather than a huge laundry list.
For PSP:
Persona 3 Portable $19.99 - I haven't played the PS2 version, but didn't think that the PSP version was lacking. I guess 3D environments a la P4G would have been cool, but the fast travel probably shaved 5 hours off of my playthrough, at least, and the direct party control is indispensable for many of the boss fights. The game has a great story, and many great characters. Not quite as polished as P4G, but still a hell of a game.
Patapon 2 $5.99 - One of my favorite games ever, and the best in the series by far. Scratched my loot, strategy, rhythm, and 2D game itches all at once. BUY IT.
Dissidia 012: Duodecim Final Fantasy $19.99 - A surprisingly well done made, 3D fighting game that pulls from all corners of the FF universe. Really fantastic, feels like a fast paced mech fighting game but with final fantasy characters.
For PS1:
Metal Gear Solid $9.99 - The grandaddy of all stealth games is still great today. Also not super long, which is a plus in my book. IMO, the best MGS available on PSP (but not Vita, that goes to 2/3).
Vagrant Story $5.99 - Final Fantasy meets Metal Gear Solid meets DnD. This game is very obtuse and sometimes punishing, but it is worth it. One of the most rewarding games I've ever played, and it has great post-game content/NG+.
Suikoden $5.99 - The prequel to the best JRPG ever made, and a great game in it's own right. Great world, story, characters, battle system, and graphics. The whole package.
For both:
Final Fantasy Tactics/War of the Lions - A really great game that you will see recommended a lot. Some annoying archaic aspects, even in the remake, but if you like SRPG, you'll still love it.
Castlevania Symphony of the Night/Dracula X Chronicles - Not my favorite metroidvania, but still a damn good one. The game looks absolutely stunning on the Vita OLED as well.
Note: I'm leaving out a lot of games that aren't available on PSN.