Oh, wow, thank you so much for the cool question, Sir or Madame, because I have actually put a serious amount of thought into this!
Well, in the first place, he’s still going to have to be in some sort of hired position for Bertie. It wouldn’t work to have them just be neighbors or friends or what-have-you, because Jeeves is a very specific, controlled creature and the way I see it, he needs a contract to do what he does. Kind of Faustian, actually.
He’d probably be a personal assistant or a caregiver.
My own idea was this: Aunt Agatha remembers a time (or remembers people who remembered a time, or perhaps has simply been watching too much Downton Abbey) when the upper classes had Ideals and used to rely on personal servants to keep them coordinated, and she’s rather depressed watching her foolish nephew cheapen his aristocratic blood by getting drunk all the time and wearing horrible stuff from Abercrombie and Fitch, so finally she hires this fellow essentially to go over to Bertie’s house and keep him in line, only to bitterly regret it when the man ends up making things even worse by encouraging Bertie in further tarnishing the family name (but at least he gets him to wear nicer clothes).
isfjmel-phleg’s idea, which I really love, was for Aunt Agatha to be the CEO of a large company which Bertie has a position in that he can’t handle on his own, so that Jeeves, his PA, is actually doing the majority of the work for him while sorting out his friends’ problems in his spare time.
Given how we human beings usually get our help these days, part of me is really tempted to make him a search engine or a Siri-like device on Bertie’s phone or otherwise not a literal person, but he could hardly control Bertie’s life in the way he does if this were so. (Besides, other people would download him and then they wouldn’t have to go through Bertie to use him, which is where 99% of the plots come from.)