Gwaihir - I side with Lottie saying that had he dropped the Ring to Mount Doom, he'd have been famous, but tough luck
Tol-in-Gaurhoth - I would have saved this as a game-pun from more appropriate tater, personally. There would certainly be more fitting taters for it.
Mordor - certainly not disagreeing.
Feanor - nor with this, though "infamous" might be even better.
Bard - I think he's so sadly neglected, poor second to Aragorn when talking of returned kings. I would give him the credit.
Great Goblin - this actually made me laugh, a "great" goblin indeed
One cannot doubt it from the title!
Shadowfax - meh, just a horse
Two Trees of Valinor - certainly. Just like Mordor or Feanor or Elrond. The problem is, maybe there are too many of those "simply famous".
The Prancing Pony - I really really like this idea. Again, in the "realistic" scale, it wasn't THAT famous (in Middle-Earth, very small percentage of people knew it - certainly less than Feanor or Two Trees), but in its area, it definitely was...
Turin - also more "infamous" than famous, I would say. I would pick Bard over him, only to give him the credit.
Glorfindel - certainly not. Sorry, but if you get kicked out on behalf of some Arwen... nowadays, his horse is possibly more famous than him! And that's a poor score.
Elrond - sure, but see above.
If I were to argue for any of these, I would support Bard the Sadly Neglected, Goblin the Great or Prancing Pony. However, thinking of it, if we wanted to be punctual, probably the most famous by the Third Age was Mordor. I mean: most of the people in Middle-Earth had at least heard about it, while even though e.g. Feanor is certainly famous, the Hobbits (!) had probably no idea who he was. Of course, in the First Age, everyone would know him. But yeah, that idea about Mordor... because imagine:
everyone, even Hobbits, or even Easterlings and Southrons would know him, whereas they would not (possibly) know about e.g. the Two Trees etc. So actually, maybe Mordor is the best.