From Mario Pei’s 1949 The Story of Language. Pei writes, “the potato, for its part, was in disrepute some centuries ago. Some Englishmen who did not fancy potatoes formed a Society for the Prevention of Unwholesome Diet. The initials of the main words in this title gave rise to spud.” Like all other pre-20th century acronymic origins, this one is false.
The word Spud comes from the digging implement used to uproot potatoes. The word is of unknown origin and was originally used as a term for a short knife or dagger. This sense dates to the 15th century. It subsequently came to be used to denote a variety of digging tools. About 1845 the name was transferred to the potato itself.
Why knives were once called spuds is unknown but the origin is not an acronym.