I think I recall someone on a TV show, maybe QI, discussing the idea that the street name was assigned by some geeky teenager on work experience at the local council . Thankfully, or sadly, depending on your outlook, this was not the case.
According to Brewer’s Dictionary of Phrase and Fable (1898), a book “Giving the Derivation, Source, or Origin of Common Phrases, Allusions, and Words That Have a Tale to Tell”:
So named from the processions of knights from the Tower to Smithfield, where tournaments were held. Leigh Hunt says the name originated in a sign or some reference to the Herald’s College in the vicinity.