You say it is working. First I would check how long the battery had been in place. If more than 3 to 4 years, I would replace immediately. Watch out for the polarity because some of the Cartier quartz movements had positive side down (facing the dial) and the battery held in place under a bridging clamp. If a clamp, one end will have a slotted screw hole while the other end also screwed to allow the clamp to swing aside. Check the open end of the clamp slides under the insulation when re-fixing, otherwise the battery will short.
Leave the crystal, dial, hands and crown well alone. They are in very good condition and timeless. I think the watch is not water resistant, unlike now for Must de Cartier, so care is needed for cleaning the case.
The case has lost a lot of its gold plating through wear. I would gently remove the silver tarnish with an impregnated proprietary silver polishing cloth and avoid using any liquid cleaning fluids. In my view the cost of gold re-plating would exceed the value of the watch. In any case the cleaned case reflects the true patina of your mother's watch, to be treasured as a keepsake. Keep the watch in box with a small sack of silica to absorb any moisture.
If to be worn, and why not, even though a small size? I would change the strap to a burgundy crocodile of a width to tightly fit the lugs. Does the now fitted strap still have the Cartier plated ardillon pin-buckle, which adds value? They can be expensive to replace.
Enjoy this very personal heirloom.
Clive