We found out that the name comes from the salt that it used to extract from the sea, using big salt pans. We still have no idea where the Budleigh bit comes from.
According to their own blurb, "Budleigh Salterton was made famous by a painting by Sir John Everett Millais. His picture ‘The Boyhood of Raleigh’ was painted from the Octagon, situated in Fore Street. The ancient wall featured in the background of the painting still stands today!" Being culturally inept, we'd never even heard of John Everett Millais or his picture ‘The Boyhood of Raleigh’, but we had heard of Budleigh Salterton.
Anyway, we came here, stopped for a photo or two, and moved on - slightly disappointed.