I'm gonna assume from what I know of you your PSP is the same as mine.
Field Runners (mini) - It's technically strategy (It's Tower Defence) but it's fun, simple. Maybe a bit too simple tbh.
Tetris (mini) -
Gran Turismo PSP - It's a pretty good game. Shiny looking. Main reason my step-dad picked up a PSP
Riff Everyday shooter is a pretty good twin stick shooter that combines with music.
There's a fair few PS1 games that are always good.
Unfortunately most of my collection is JRPGs, which might not be suitable.
If he's into JRPGs, then I'd recommend Disgea. Very grindy, but fun and simple enough.
And most PS1 and remade PSP Final Fantasy games.