How did Saruman know who Gimli was ?

[Rumil]

Agree that Saruman probably heard of Gimli via Wormtongue.

However I'd guess that he had some interest in the Dwarves of the Lonely Mountain. After all the Battle of Five Armies was a really major event that altered the balance of power somewhat. I think Saruman would have been rather fascinated as to how the Grey Wanderer somehow managed to arrange the demise of Smaug the Magnificent.

More importantly, putting 2 and 2 together, Balin's occupation of Moria might have made him sit up and take notice. Saruman was the expert on Rings of Power, I bet he had figured out that there was still one of the seven unaccounted for, though I'm not sure if Gandalf let on that it was taken in Dol Guldur? If not, then Saruman would have loved to get his hands on it and Moria wasn't too far for his spies to travel, or to do a bit of Palantir-ing.

Gimli was probably one of the top few dozen Dwarves of Erebor (in terms of social status) and Saruman, being the political operator we know he was, might well have complied information on the top players amongst the major powers of the time.