This is a Clement 1903 and, judging from the late serial number, is probably chambered in .25 ACP (6.35mm) instead of the earlier 5mm Charola y Anitua caliber. The caliber transition occurred when the FN 1905 vest pocket pistol, using the newly designed .25 ACP cartridge, entered the market c. 1905/06. Given the popularity of the FN 1905, Clement saw the writing on the wall and realized that the .25 was the future. Thereafter, all Clements were .25s (well, the smallest ones, at least - there were .32 ACP Clements as well).
In excellent condition, this gun has considerable collector appeal. The bad news is that your specimen, given the photos and description, seems to leave a bit to be desired regarding condition. It's certainly not bad, though - especially since the original grips are not cracked or chipped! That's impressive for a gun that's over 100 years old and one that, from your story, saw some use in your grandfather's hands.
It also looks as though it has a very decent amount of the original blue remaining. That's always a good thing.
Depending on what parts are missing, whether the gun is basically functional and the condition of whatever internals remain, value could range considerably. If forced to guess, I would estimate perhaps $200.00 to $400.00.
ATTENTION: If you do decide to disassemble it, show this venerable old gun some love by using the PROPER TOOLS FOR THE JOB! Those screws should only be loosened using gunsmithing screwdrivers with parallel-ground bits - NOT regular screwdrivers. There's nothing more saddening and exasperating than seeing stripped screw heads on old guns simply because the owner couldn't be bothered to do it right. The screw heads on your Clement are in very decent shape - please keep them that way!!!