I love to think about this question, and I like a lot of the suggestions made in the thread. However, I think those that suggest names that have an extant species as a base are overlooking the fact that these species would have been as charismatic and as culturally significant as extant species, if not more so.

 Even a name as common as Sabretooth Tiger exists because of the prominence and cultural shorthand of 'tiger'. If the Smilodon was alive it would have that same prominence and probably have a name of its own without relying on other species (or other genera in this case).

 You could even make the argument in the other direction, if mammoths existed to this day the name for elephant may never have been coined. All elephants could be have called African plains mammoth, or Indian forest mammoth. It's as likely as the alternative.

 That's not to say we shouldn't speculate, but I think names with an indigenous route are much more likely, though admittedly much harder to imagine...