How did Saruman know who Gimli was ?

[Legate of Amon Lanc]

I second Inzil. Also, apart from that, Bilbo's connection to the Ring was unknown to Saruman basically until the very end, when he got straight to the source - i.e. Gandalf, whom he imprisoned in Orthanc. If he would be interested in learning how did the Ring end up in Bilbo's possession, it would have had to be solved then, and again, we are basically around the same time there. Anyway, learning the Dwarves' names would have been of little importance to Saruman. I mean, Balin, Dwalin, Fili, Kili, Dori, Nori, Ori, Oin, Gloin, Bifur, Bofur, Bombur and Thorin would sound as funny to Saruman as they do to a reader of the Hobbit, except maybe for Thorin who was of some renown, but as good as dead (well, rather dead, actually). These names were of no use to Saruman as they didn't have anything to do with the Ring, and like I said above, at the time when Saruman could have learned about their connection to Bilbo, his immediate concern would be to get the Ring itself, which was somewhere in Eriador being chased around by the Black Riders.

AT MOST, though highly unlikely, Saruman could have - out of interest, or rather out of the pedantic obsession to know everything about the Rings - "googled" the names of the thirteen Dwarves around the time of Gandalf's imprisonment in Orthanc, which could have coincided interestingly with Gimli and Glóin's journey from Erebor to Rivendell, and maybe then Gimli's appearance in the Fellowship might have been more of notice to him, as he would get the feeling that "something is going on here" as the recurring characters revolving around the Ring appear. But personally I somewhat doubt that.