Palm Sunday is so called from the custom of eating figs on that day. The practice arose from the Bible story of Zaccheus, who climbed up into a fig-tree to see Jesus.
Many other festivals have their special foods; as, Michaelmas goose, Christmas, plum-pudding, Shrove Tuesday, pancake day; Ash Wednesday, salt cod; Good Friday, hot cross-buns; pasch-eggs, roast-chestnuts, etc., have their special days.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Fig Sunday from Infoplease: