The most common problems to check for...
1) If you've got one, take along a generic, one-contact flash to make sure the hot shoe works. If it's got a PC cord, also try the PC socket.
2) The MX is prone to developing a short in the light meter contacts. This means that you can put a brand new battery in and discover a few hours or days later that the battery is dead. This can happen even with the shutter button lock locked. You can try checking for this by looking through the viewfinder and pressing down on the shutter button while it's locked; if the LEDs in the display come on anyway, it has the short. There's some info here:
Just skip down to the last two paragraphs.
3) The MX is prone to dents on the prism housing. This doesn't necessarily mean there's anything wrong with it, but it's something to look for.