I care a lot about video game characters, to be honest. It's what gives life to a game, some of the best games out there demand that your characters are developed and to some extent fresh.
I might get some negativity for this opinion, but I was actually particularly fond of Tidus (FFX). Sure, he's got that whiny attitude and stuff, but he was young and he was just trying to get back to Zanarkand. To me, it felt real. Certain scenes seemed a bit forced and were embarrassing (laughing scene, I'm looking at you) but FFX is my favourite FF game (played them all except XII) because of the characters and the story.
Same goes for many other games. Half-Life 2, as mentioned so very often, is one of the best games when it comes to personality, fleshed out characters, ect. I don't think I need to say much more.