I stumbled across this Quora post, and it turns out that Edward M McCreight actually answered this question 6 years ago, at CPM 2013. In this video from the conference, starting at the 16 minute mark, someone asks him what the "B" stands for.
The tl;dr is that the B doesn t really stand for anything, and that team that worked on the B-Tree just came up with the name B-Tree over lunch.
Based on his answer, it sounds like they either came up the name "B-Tree" and then tried to figure out what the "B" stood for afterwards, or they came up with a few possible names(Boeing, Balance or Bayer (the senior author)), but they never decided on which one to go with. Since each of the name ideas started with "B", they then went with the name "B-Tree".
The quora answer provides a written transcript of his answer, for those not wishing to watch the video (or should the video ever be removed):
You just have no idea what a lunchtime conversation can turn into. So there we were, [indistinct] and I, at lunch , we had to give the thing a name. And we were, so, B, we were thinking… B is, you know… We were working for Boeing at the time, we couldn t use the name without talking to the lawyers. So, there is a B. It has to do with balance, another B. Bayer was the senior author, who did have several years older than I am and had many more publications than I did. So there is another B. And so, at the lunch table we never did resolve whether there was one of those that made more sense than the rest. What really lies to say is: the more you think about what the B in B-trees means, the better you understand B-trees.