I totally agree that every detail of the Erebor quest should be of a great interest for Saruman. Smaug's ellimination and the Battle of Five Armies were very important events; he cuould also have felt there was something dodgy in that, though the Ring was kept a secret. So Saruman must have been be aware of Gloin's name.
I don't think he knew anything about Gimli before the Fellowship departed from Rivendail. However he could have learnt it either from his spies in Hollin (there could be not only animals, but Dunlendings), or from Wormtongue, or himself in Farngorn, as have already been said here, so there was no mistake.
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