My suggested wording, @JW: "At the start of each of your turns that this remains in play, choose one: Exile a card from your hand; or Exile this for +3 Cards."
I thought about this, but it is unfortunately self-referential. When you decide whether to discard this during cleanup, you need to ask "will this do something next turn?", and if not, discard it. But now it does something if it stays in play and doesn't do anything if it doesn't stay in play, so you can make the argument that it should be discarded at the end of the turn.
What about Gubump's wording, plus "(This is not discarded during clean-up)"?
One issue with this is that words in parentheses generally clarify the rules, they don't modify them (meaning the card would function the same with or without them). I don't think that is what is happening here, at least not clearly.
I would suggest something different. I think the closest official example is Crypt, which is the only duration that does not remain in effect (and therefore, barring rare cases of trashing, in play) for either a fixed number of turns or the rest of the game. The duration effect of that card says "While any remain, at the start of each of your turns, put one of them into your hand." We can generalize this to "[Condition], [timing], [effect]." Here, the Condition is really "until you Exile this" or "until you choose the second option. The timing is the same, and your text for the effect is already fine (except that Exile and Cards should be capitalized, and the choices should be separated by a semicolon). Thus, I would word it as follows:
Until after you Exile this, at the start of each of your turns, choose one: Exile a card from your hand; or Exile this for +3 Cards.
The only issue with this is that if you Trash it with Bonfire, you would be stuck Exiling a card for the rest of the game. Given that that would be entirely a player's choice, I do not see that as a huge problem. It does become a problem if another player uses Possession and then trashes it with Bonfire, forcing you to keep Exiling (unless you played it again and then Exiled it, or Exiled it from your hand, although that creates tracking issues if you have multiple copies).
You might solve that by saying:
Until after you Exile or trash this, at the start of each of your turns, choose one: Exile a card from your hand; or Exile this for +3 Cards.
I'm not sure if you need to add that for such a fringe case, although it doesn't really add that much verbiage. Another solution could be to use another card for tracking to avoid any self-reference. That might look like this:
Gain a Silver. Set it aside. While it remains set aside, at the start of each of your turns, choose one: Exile a card from your hand; or discard the set aside Silver and Exile this for +3 Cards.
(I think Silver fits thematically; the Gallery gains you Silvers like the Masterpieces it displays). The obviously adds an entirely new element of gaining you a Silver. If the gaining gets monkied with, the Silver won't be set aside (under the stop moving rule), and the duration effect will fail, but I don't think that's a problem, since it will (I believe), always be the choice of that player. At $6, I don't think Silver gaining will be particularly strong, but given the card's powerful Exiling I don't think it hurts to much either. If Gallery gets trashed by Bonfire, the player still has the option to discard that Silver and end the effect. All that being said, this does add quite a bit more text, a whole new element/moving part, and complexity, all of which are probably needless because option (b) probably works fine (unless you want to add the element of gaining the Silver).