[rossdavidh]

The only right answer. If they cannot accept a diplomatic, reasoned and carefully explained "we can't know, but I'll finish as quickly as I can" answer, then they probably won't be very understandable when things turn out to take longer than expected. Or if they change their mind about what they want and expect you to nonetheless finish on the original schedule because it's just a small change. Politely and gently refusing to provide an estimate for the unknowable, is a good way to avoid working with clients who will be not worth working with anyway.

Hard as this is up front, it's harder to explain afterwards why your estimate didn't work out, and it's harder to eat the cost difference yourself. Be gentle and polite, and accept that they may or may not be willing to work with you on trust that you will do your best to finish as quickly as you can.