In Chapter Nine, Bud is sitting in the library when he recalls a conversation that he had while being bullied by Billie Burns. Billie had bet a nickel that Bud didn't know who his father was, when Bud announced, "...[M]y daddy plays a giant fiddle and his name is Herman E. Calloway."
The small idea (his father's identity) stars growing when Bud thinks about why his mother had kept all the flyers of Herman while never speaking of Bud's father. Bud feels that this "little idea had gone and sneaked itself into being a mighty maple."
Thus, the big idea that Bud is referring to is his plan to head west to Grand Rapids in order to find the man he believes to be his father... that is to say, Herman E. Calloway!