The Private Correspondence of David Garrick (vol. 2, p. 212 on googlebooks), has a letter from one G. Butts to David Garrick which includes this passage:

And now, Sir, I have concluded my dramatic history, which naturally enough reminds me to mention, that the person who is principally responsible for my violation of the chaste Melpomene is a friend, whose friendship for me is my richest possession, and hereafter may be my greatest honor. His name is Woodhouse, (and we too “were nursed upon the self-same hill,” that very hill on which with a deliberate falsity I have roundly asserted that you were nursed;) his father was the late Dr. Woodhouse, of Litchfield; he is well known to your friend the Archbishop of York, to whom I refer you for his character, as you would receive it from me with a suspicion of some friendly garnishing. (Emphasis added; original punctuation left intact.)

Apparently, from what I see in the rest of the letter, Butts was a writer, perhaps a playwright. The "chaste Melpomene" is the muse of tragedy, which suggests the latter. In this letter, Butts is essentially telling Garrick how he got into the writing game, as well as some of his thoughts on various plays (he mentions Lear), etc. I think he is engaging in some self-deprecation with his comment on Melpomene, that his efforts may not be the best but they are the best he can do.

What all this means for JA's choice of Garrick's riddle I do not know. Perhaps she had already chosen Woodhouse as the family name in Emma and someone suggested the Garrick connection. That's just conjecture, but I figure there is some reason she threw in the Garrick riddle and that reason is as good as any I've heard.