What is the most important choice a character makes in Mark Twain's The Prince and the Pauper? Why does the character make that choice?

There's a mutual choice that's the most important for the story to happen at all, and that is that the two boys change clothes and positions.
If you really want the most important choice by a single character, it would be the choice to read, which is Tom's choice. This is described in Chapter 2, where Twain writes, "He often read the priest's old books and got him to explain and enlarge upon them. His dreamings and readings worked certain changes in him, by-and-by."
As far as why he did this, there are several reasons. The first is, of course, that he was a poor beggar boy. His actual day-to-day life has little going for it, so he reads to escape his reality. He is open to reading because Father Andrew tells him stories to make him feel better and entertain him—and he did this because Tom was a poor beggar.
This was the most important choice because it starts the change in Tom. He moves from being passive to being active, and he reshapes the beggar world around him into a royal court.
This sets up the later plot in which the boys change places.