The implication is that the watch, which would have contained steel parts and that had been left for centuries in an unsealed cave, was rusted or decayed beyond reasonable repair and Clemens simply left it to be disposed of or kept as a curio by the crew.
Clemens notices something on a nearby table... picks up the watch from the cavern.
CLEMENS: My watch...
GEORDI: It was found in the cavern... near Data's head.
After five hundred years, I doubt that it'll work either.
Clemens puts the watch back and regards the lifeless android sadly.
CLEMENS: Mister Data... I fear I sadly misjudged you.
As I have misjudged many things
Times Arrow, Part II - Original Screenplay
In real life Clemens (Twain) is known to have owned multiple pocket-watches. There are several numbered among his stored personal effects on display at the Mark Twain Museum and at least two on current display.
It might also interest you to learn that he gave his name to a shady pocket-watch mail order company that marketed a patented "Mark Twain" watch (that stopped regularly and failed to keep correct time). After the failure of this pyramid scheme, he was utterly scathing about watchmakers as a profession.