The story
Possibly baby talk forms of Humphrey, since Dump and Dumphrey have been recorded as pet forms of this given name. Various suggestions have been offered for the original meaning of the nursery rhyme, chiefly concerning the English Civil War, although none has received general acceptance- modern scholars believe it may have been intended to be a riddle, with the answer of "an egg".
Etymology from Wiktionary, CC BY-SA 4.0.