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.

[CSteefel]

It is a bit hard to imagine that Saruman's animal spies were able to pick up that the dwarf's name was Gimli and that he was son of Gloin. Even the night time visit to the the campfire would not have given him this information.

Only one I can think of is Wormtongue, who would presumably have heard the name from Eomer or one of his men, since this is where Gimli announces that he is Gloin's son. Since Eomer lent them the horses, and the guards at the gates of Edoras knew that they were returning with those horses (as Gandalf notes upon arrival), the names of the three wanderers who they had met to the north would be known to Wormtongue...

[Inziladun]

Originally Posted by CSteefel

Only one I can think of is Wormtongue, who would presumably have heard the name from Eomer or one of his men, since this is where Gimli announces that he is Gloin's son. Since Eomer lent them the horses, and the guards at the gates of Edoras knew that they were returning with those horses (as Gandalf notes upon arrival), the names of the three wanderers who they had met to the north would be known to Wormtongue...

Now that I think about it, that's probably the most likely. Éomer did indeed hear Gimli say he was the son of Glóin at their first meeting, and that was the only time Gimli's father is mentioned in that part of the book. At the time Aragorn, Legolas, and Gimli see the old man at the edge of the trees, only Gimli's name had been said when the old man was likely to have been watching.