I think Wilberforce is just one of those names Wodehouse thought was funny! He had a good number of those, names that just got stuck in his head that he liked to use repeatedly or that he associated with certain archetypes, like how he thought "Meadowes" was a quintessential butler name (and we saw it used by Catsmeat when he was posing as a valet). I find it quite amusing, it's like how - at least when I was a kid - kids thought that "Bob" was the funniest name there is, and whenever someone wanted to come up with a funny name for a character they would name the character Bob (from what I've heard, the child-certified Funniest Name is apparently now Jeffrey). It has that kind of gleeful energy.
Anyway, "Wilberforce" comes up quite a bit in Wodehouse's work - or at least, it's come up at least twice in a non-Bertie context in the work of his I've read. The valet of Bertie's prototype, Reggie Pepper, in the story "Concealed Art," was named Wilberforce, so it’s quite funny that Bertie ended up with proto-Jeeves’ name and Jeeves ended up with proto-Bertie’s name. And then I ran a concordance for all instances of "Wilberforce" in the Jeeves books, and we can indeed see that it's never used until volume 3 of the omnibus:
The books in the omnibus aren't in publication order, and the last book in volume 3 is actually Very Good, Jeeves, meaning that the first instance of "Wilberforce" is as a last name for the family of the waitress Jeeves was trying to set his friend up with in "Indian Summer of an Uncle." Second usage is in The Mating Season, which is where it's revealed as Bertie's middle name. But then it comes up again in Ring for Jeeves, where it isn't even anybody's name, but someone is trying to remember a different W name and draws "Wilberforce" first. Schoolkid gigglingly naming his googly-eyed sock puppet Bob, I'm telling you.
(ETA: wait, no, actually, I tell a lie, the first use of Wilberforce was in “Jeeves and the Old School Chum” where it was the name of Bingo’s uncle. Missed that one. That’s funny, I always assumed when I read that story that this was the same uncle that appeared in The Inimitable Jeeves, but no, that one was Uncle Mortimer)
And then, of course, we have the last usage in Much Obliged, Jeeves, where Aunt Dahlia expresses her sympathy for Bertie having been given a comedy name as his middle name.
I think it’s likely that Wodehouse didn't decide on that bit of canon until The Mating Season; if he'd known it all along, after all, one might have expected Bertie to say something about the people in "Indian Summer of an Uncle" having a last name that's the same as his middle name. It is nice to see Wodehouse remember it in MOJ! And not just the "Voice That Breathed O'er Eden" bit, but also that Bertie only uses his middle name in moments of high emotion (of course, we could chalk this up to the fact that while Wodehouse IS notorious for forgetting his own canon, he's also notorious for REPEATING his own canon, or just repeating lines or jokes he thought were funny the first time. Why let the fact that you've already told a good joke get in the way of telling it again?)
(This was all kind of a surprise to me too, because I would have SWORN I could recall some instances of him calling himself B. W. W. or Bertram W. Wooster or whatever, but I can't find anything for those no matter what way I search, with spaces or without, etc. Have I only seen it in fanfic, then? I guess this is one of those cases where a particular piece of fanon has been adopted so heavily that everyone forgets it isn't canon!)
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