Arthur Wellesley was born Arthur Wesley, and changed his named on his rise to fame and fortune, deeming Wellesley to be a bit nicer, more upper class. (I don't know why anyone would ditch the name Wesley...). He was a hero to the people of England, and moved in the very highest social circles, mixing with all the royalty and whatnot.
Soldiers being a superstitious lot, many myths and superstitions were attached to the Duke. He named his battles after the town he stayed at the previous night (hence the battle of Waterloo was not actually at Waterloo, but he did sleep there). It was believed by his soldiers that it always rained before his battles. This was called "Wellington Weather". That's why wet-weather boots are called Wellingtons.