S&W referred to that gun as a .32 Hand Ejector, Second Model. Collectors have distinguished five different sub sections in that classification, and this would be called a fifth change gun.
The serial number points to shipment about 1911/1912, so the gun is a century old.
The gun is probably more valuable as an heirloom than as trade goods. Prewar .32 HEs can be had for as little as $250 when they are worn or a little loose, but most would probably be priced in the $300-400 range unless they were in almost new condition and still in their original box. Those would be worth several hundred dollars and might even approach a thousand.
Yours looks to be in semi-good shape for a nickel gun, and I'd guess a value around $350 if you wanted to insure it. Maybe a little more, but demand for these old .32s has fluctuated a lot over the years.
There is never any harm in cleaning or lubricating a gun if you do it properly. Some cleaners have solvents that attack the nickel substrate and will actually take nickel off, so make sure you use an oil that is safe for that kind of finish.
ADED: Guy got in first while I was dithering. I look at prices with California eyes, so my estimates can be higher than those originating in other states.
