When responding to "Who sees the hammer?" When the person who sees the hammer answers the questions. Should they use "me" or "I do"?
"I do" is correct in terms of proper formal grammar, but "me" would be commonly used and not generally thought incorrect. For some audiences, you may find you are expected to use "I do".
They should say “I do.” Think about replacing “who” in the question with a pronoun to turn it into a statement. The correct sentence is “I see the hammer,” not “Me see the hammer.”
EDIT: see comments below. I realize now that this answer is in fact specious, as another commenter pointed out. "Me" is very informal but not incorrect when it stands on its own as an answer to the question. I am leaving my original answer above so it might be used as a learning opportunity for future users of this StackExchange community.