Alright, let's have a go at this. I'll try and choose characters who haven't already been picked:
Atrus - Myst series
"I realized the moment I fell into the fissure, that the book would not be destroyed as I had planned."Theme song - Atrus' Theme, by Robyn MillerMyst has some incredibly deep story if you are willing to read all the journals in the games, and the novels as well. Atrus, played by Myst creator Rand Miller himself, is the central character of most of the series. You learn about his childhood, his family, the worlds he has created, his mistakes and his regrets. Atrus is the central figure that connects all the plot points in Myst.
Kyle Hyde - Hotel Dusk series
"Maybe I can deduce my fist to your face. Bet that would shut you up."No specific theme song, here's the full soundtrack insteadKyle Hyde is a really cool character, in every definition of the word. The games are set in the 80's and ooze noir and hardboiled detective mystery from every possible outlet. Hyde himself is a former detective, now a travelling salesman, though that's just a cover to move around a lot and investigate stuff without being noticed. He's chasing his former partner, who disappeared after an undercover job went wrong. He's usually calm and collected, but he can get quite emotional if provoked. Whether he's helping a child solve a jigsaw puzzle, grilling a suspicious person for their hidden secrets, or drinking a whiskey on the rocks, Kyle Hyde does everything in the coolest possible way.
Lynne - Ghost Trick
"Oh, I just LOVE stomping out evil! It makes me feel so alive!!!" (said while dead)
Theme song - Lynne ~ A Targeted Redhead, by Masakazu SugimoriLynne is quite a light-hearted and energetic character who you initially assume will play a "hyperactive sidekick" type of role in Ghost Trick. She has a lot of amusing quotes and she's generally a really fun character. However, as the story unfolds, you start to discover that her motivations aren't as simple as first assumed, and she has quite a dark past that is revealed over the course of the game.
Mordin Solus - Mass Effect
"Had to be me. Someone else might have gotten it wrong."Theme song - Mordin, by Jack WallGarrus is a cool guy and all, but Mordin has a more interesting backstory, in my opinion. He's basically a badass special forces guy as well as a super smart scientist, which is a pretty interesting combination. He's also quite morally ambiguous, which makes for some interesting dialogue in places, especially since most of his sidequests involve his participation in the krogan genophage.
Miles Edgeworth - Ace Attorney series
"I object! That was... objectionable!"Theme song - Great Revival ~ Miles Edgeworth, by Akemi KimuraIn the first Ace Attorney game, Miles Edgeworth is an antagonist who you immediately dislike, but over the course of the trilogy, his character changes quite a lot. He goes from an incredibly arrogant and aggressive prosecutor who will do anything to win, to somebody who believes in using legitimate methods to persue the truth, even if that truth ends up being something he doesn't like.
What's great about Edgeworth as a character is that he's multi-dimensional. He prides himself in being a classy and refined person, but he's not perfect - he can get flustered like anyone else and can even get miserable and sulky at times. He'll sometimes get surprised by something unexpected and panic in court, resulting in him choking on his words and saying something silly. He's also a closet fan of the children's show
The Steel Samurai, but tries as hard as he can to hide the fact, usually resulting in amusing dialogue.
His dynamic with detective Dick Gumshoe has a perfect combination of fondness and disdain. In the chapter you play as Edgeworth and in the two Edgeworth spinoff games, they make for a very amusing pair as they perform investigations, with Gumshoe playing the funny man and Edgeworth as the straight man.