Steambot Chronicles - This was a interesting gem from the PS2. The plot is a bit hard to explain, but the start beings with (surprise surprise), a main character with amnesia who winds up playing in a band with a couple mecha fights along the way. A interesting bit they've done with the character is the dialogue options. You could make him a selfish ass who's money-oriented, nice but cowardly, and other just with the various lines. And there's quite a lot to do in the game. There's of course, the whole playing a instrument (each one having a different mini-game), being a make-shift taxi, being a professional mecha arena fighter, and other random things to deter you off the main story. As for customizing your titular vehicle, there are a ton of parts to make it suitable to whatever you'd like.
On the other hand, the game is pretty clunky, especially when it comes to the controls. There was going to be a sequel, but sadly, it doesn't seem like it'll come to light any time soon.