This is a tough one since I've been playing two thief characters recently.
Shade-In-The-Shadow is a filthy lizard. Even though an honorable Morag Tong assassin, this Argonian's ultimate goal is to free all the slaves in Vvardenfell. While at it, he steals whatever he likes to steal, whenever he wants.
Aerruil the Eloquent, though, is a very unorthodox Bosmer. While people of that race are mostly looked down as elves, he is actually a high society mover and shaker. A Hlaalu noble. He granted his position very quickly with his natural charms and unusual near-man looks. Charismatic and entertaining, people forget this young and attractive guy is "just" a Bosmer. There are nothing more than wild guesses linking him to another surprising new force in Vvardenfell, the mysterious thefts that sometimes occure even to most influential people. Those thefts are done with utmost care and completely unnoticed. It's often difficult for these robbed people to notice that they even are robbed - usually just one remarkable item is missing from their collections. Everything else that was there has been left completely untouched.
So yeah, he's a stereotypic for a gentleman thief, only robbing what he feels like will be a good, valuable memory from his successful robbery. He's not actually stolen that much yet. While I feel I'm at certain point with him, I begin to leave a Heather at the crime scene, for a visiting card.
Even though I've played Shade a bit more, I like Aerruil even more. He has a better gimmick or something. Shade has an agenda: to free the slaves, often of his own race, and other underpriviledged as well. And he believes in disposing of (bad) folks and even taking honor from it. But Aerruil is so typical attractive gentleman thief from literature and films, and mostly a thief, after all, unlike Shade, so I'm going with him in my answer: I only steal certain items. The more unique the better of course.
Uniqueness isn't the only factor though. Importance, value, appearance, others count too.