To summarize the summary of the summary - he came up with better ideas, or the story had to change for different media.
The common thread between the first two versions of the plan is Benjy and Frankie Mouse give up almost immediately on retrieving The Question from Arthur's mind, opting instead for making something up, like "How many roads must a man walk down?" That's really the joke - after a ten million years plan goes bust, they opt for manufacturing a solution, which is what people have been doing all along.
The original radio play is quite an embryonic thing, and quite a number of things that we see as canon were added in later. Similarly, a lot of jokes vanished - the Haggunenons apparently evolved themselves right out of other media and adaptations (possibly because they'd be so expensive to show).
So in the original radio play, the mice offer Arthur a ridiculous amount of money if he should happen to work out The Question. Funny, but it's conflict that drives a story. So in the book, the re-recording of the original play, and the TV series, while money is still offered, Arthur's brain will have to be removed, and properly treated(*) to extract the question. They offer the suggestion of replacing it with a (simple) electronic one, which would be programmed not to notice that it was a replacement.
By the time the film came along, they upped the ante again - it was clear that the mice never actually intended to part with any money, as they drug the really lavish meal they offer Arthur and his team, and get right to trying to extract (with extreme prejudice) the information, and its container. That goes a bit too far - the suggestion of the action is funnier than actually seeing it about to occur. Also, the "Aw heck, let's make something up" joke is lost.
Each change made to the plotting for the various media (save for the movie which was completed after Adams' passing, but supposedly with his approval) was simply Adams coming up with what he thought was a better idea, or a necessary change for the larger narrative to fit the new media. The plotting of the original six episodes of the play changed quite a bit when the first two books were written. This was so the first book could have a relatively happy ending - they escape Magrathea, and only go to Milliways in the second book, which is now located on Frogstar.
Adams never worried much about continuity - if a better idea came to him, he'd use it, and not worry terribly much about how it might contradicted past works. IMHO, most writers do this, he was just much more blatant about the not worrying part.