How did Saruman know who Gimli was ?

[Legate of Amon Lanc]

Originally Posted by Inziladun

Courageous he certainly was. However, I think he would have been of absolutely no interest to Saruman were it not for his involvement with the Company of the Ring.

Exactly. With all respect to Gimli, son of Glóin, I am pretty sure Saruman's attitude would be "Dwarves? These funny little ground-diggers? Does that even exist still?" (Funny, by the way, imagining that from former Maia of Aulë. But I could indeed think it was like that.)
Judging from Saruman's speech to Gandalf:

The Elder Days are gone. The Middle Days are passing. The Younger Days are beginning. The time of the Elves is over, but our time is at hand: the world of Men, which we must rule.

I think for Saruman the Elves were done for, but the Dwarves were never even worth mentioning. I mean, really. Try to think like Saruman: he would look at the achievements in Middle-Earth purely from the basic look at the timeline. Who was in Saruman's eyes "cool"? Likely some great Elven kings of old, who had the power to overthrow Morgoth or Sauron (although most of them still died, losers them). Dwarves never really made a big difference, not in the way noticeable by Saruman. The only thing they would be useful for, if you asked, would have been probably mining gold, so that you could make nice rings out of it. And that's probably all.

So all in all, no. Gimli son of Glóin was familiar to Saruman because he happened to be a member of the Fellowship. Hm, speaking of this, I think I know - if Saruman had spies (birds and such), they probably told him how many people are in the Company and who they are, and of course he would then search for who exactly they might be. And the journey of Gimli and Glóin to the Council would have been easy to track for his agents, most likely.