Adding an overarching plot may be a good place to start.
Make it feel like there is a story with some progression to it, as opposed to a random string of robberies that don't have any end to them. The important thing is to make sure your character feels a sense of progression, and like he is being challenged and growing. In the end, it probably won't be that different from the structure of any old tabletop game.
Given that this is a solo campaign, you have a lot of opportunity to really develop some NPC characters for him to interact with. I'd create some reoccuring characters that he could come to know and appreciate.
A sample plot:
I really think you should run with him wanting to be the greatest thief of all time.
Perhaps a thieves guild takes note of his exploits after he robs a particularly notable establishment. They can then attempt to recruit him to their ranks. Because you said the player wants to be "the greatest thief of all time," maybe the story could be able him slowly climbing the ranks in the guild via going under more and more perilous missions.
Some obvious antagonists could be the authorities of whatever civilization who could eventually put wanted posters out for him. You could even throw an Inspector Zenigata type person after him who makes it their mission to catch him and throw him in jail for good. Having some higher ranking members of the guild who want to protect their seats of power within their underground society would work, too.
Some of the earlier missions that the guild sends him on could be his initiation rites to prove that he really has the moxie to be part of such a group. Later on, he could be sent on some group missions to steal some really valuable stuff that is highly guarded. When he's really well know in the guild, you could even put him on a mission where he has to steal something before a rival in the thieves guild does themselves! At the very end of the story, he could sit high and mighty as the leader of the guild and properly feel like he's earned that title.
Playing through this he'll feel as though he's really earned the title, and not like his character started out as that. Good stories have growth and struggle in them, and if you create some NPC who directly challenge him and force him to grow, he'll remember them as fantastic characters in your story.