I cleaned the contacts on my 4 button thermostat today, and shot a few pictures of the process. To get the unit off the wall bracket, use a knife or small screwdriver to pry down the tab on the bottom, then pull the bottom away from the wall. When the bottom is loose, lift off the top catch. Then you can unplug the phone type cord from the thermostat. The picture below shows the bottom locking tab that must be pulled down to release the thermostat.

Remove the 4 screws from the back of the thermostat.

Place a small amount of contact cleaner on each switch assembly, apply so that it runs down the shaft of the push button (where the screwdriver is pointing).

After applying the contact cleaner, work each switch assembly several times to insure the contact cleaner gets to all the contacts. Allow to dry for a few minutes before reassembly.
If cleaning the contacts does not cure the flaky operation, try straightening the contact legs. The contact legs are inside the blue part of the assembly. To remove them, carefully pry the small white lock (on each side of the blue box) away so the white part can be pulled up. Be very careful as the spring inside can be easily lost.
The contacts may also be turned around so the other side becomes the working set. This will become obvious after disassembly.

I've been keeping this thermostat working for many years by following these steps. Of course, if the circuit board dies, all bets are off.
Fred