Very old mirror, how to refinish?

I would advise you to leave it alone yourself. It is likely to contain mercury, used in the mirror process that long ago. This could easily be renewed by a good glass/glazing firm but requires specialist materials to strip and re-silver. They would remove the timber backing if you can only get it to them in this manner. You would have to accept that any surface scratches present already are likely to become more highlighted when finished, because of the fresh silver and todays quality of silver brightness is much better than of old.There is also a small possibility of the glass being stained (in patches, maybe not at all) on the rear suface over the years which may not be obvious at the moment but may be more evident after the process, this is a fault that cannot be rectified therefore you take your chances On balance if you can afford it I would still go for it and if not, learn to live with as it is as there is still a certain amount of charm in that state (some interior designers would give their right arm for a mirror in that condition). You could phone around your local glaziers first to ask if they offer this service, give them the overall size and mention it is likely to have been made in 3/8" (10mm) thick plate glass and it has bevelled edges and has some brilliant cut detail to the rear (this process can add a little). Tell them it is in a frame that you need them to remove it from, etc. They should be able to price the strip and re-silver part....but may need to see it in it's frame to see how complicated it is to extract before pricing that part. There is nothing you can do yourself to improve/renew, etc.

resilvering mirrors is very common and not very expensive, there is no DIY way of doing it