Honeywell Mercury Themostat 3 Wire, replace with Digital/Programmable?
What your options for replacement are (easily) depends on what your wiring is.

I find it odd that you claim it's getting inaccurate, but perhaps they built them better 50 years ago than 20 - all the 50-60 year old ones I have work as good as new, but those are the "classic round" version. They actually are subtly adjustable if needed, though non-subtle attempts can screw them up. They are sidelined at the moment, but in reserve in case of need.

Anyway, for a two wire setup the easy digital/programmable solution is to get a digital/programmable that uses batteries - I have a couple of examples of these in my view between home and work, and the older one eats batteries a lot slower than the newer one. 

The fancy/expensive/prone to becoming obsolete so you need a new expensive - "Smart" thermostats (or hacking portals into your house) generally require a wire upgrade to supply the power they want. I'm obviously not a fan, but many people do seem to love them. Some of those people are not living in your house...