Focusing on the "tired" part:
* "Tired" could be replaced with "tyred" (which is an archaic spelling of "tired"). This wouldn't change the meaning, but it would create a homophone with "tyred" (meaning "equipped with tires").
* "Tired" could be replaced with "weary." This is a synonym for "tired" and creates a homophone with "wary."
Focusing on the "root vegetable" part:
* "Root vegetable" is quite specific, but you could play on the "vegetable" part and say "vegatable" (which is a misspelling of "vegetable"). This creates a homophone with "vegetable."
Making a humorous pun:
* You could play on the idea of a "tired" vegetable being "wired" and say "wired root vegetable." This is not a true homophone but uses a play on words.
Ultimately, the best "homophone" depends on the context and the type of wordplay you're aiming for!