The riddle
What Do You Call a Deer with No Eyes?
No eye deer, no idea
Say the answer quickly: "no eye deer". It lands on the phrase "no idea". Merriam-Webster gives idea as eye-DEE-uh and deer as DEER, so the stress falls on the same syllable in both. The first two syllables, "no eye", are identical. The difference is the end, where idea finishes with a soft uh and deer with an r. The match is close, though not perfect, and it works best said fast.
What deer used to mean
Online Etymology Dictionary says deer comes from an Old English word that meant an animal of any kind, and that it narrowed to the antlered animal over the Middle Ages. Merriam-Webster notes that the plural is the same as the singular, so one deer and a hundred deer use the same word. Dear, meaning loved or costly, is a separate word that sounds identical, and the two are the classic homophones for a word game.
Dear and deer
The animal's name sounds exactly like dear, and older riddle books used the pair. A 1903 collection, Conundrums, Riddles and Puzzles by Dean Rivers, asks what animal drops from the clouds and answers "The rain, dear (reindeer)."
Keeping the deer eyeless
In the joke the deer simply has none, the way a cartoon character might lack a nose. The question exists to deliver the answer, not to describe the animal. Ask it with a straight face and pause before the last word.