It seems that any elf could bestow this title.
Frodo is named as an Elf-friend by Gildor Inglorion, when he encounters a group of elves on the borders of the Shire on his way to Bree. Gildor does not appear to have any special status among the elves.
In any case, the title seems to be largely honourary (as James Khoury says in comments). At best, other elves who heard that a certain mortal was an elf-friend might be somewhat more welcoming towards that mortal. At worst, it wouldn't make much difference at all. When the Elves of Lothlórien meet the Fellowship they are distinctly suspicious, even though it contains an actual elf and at least two Elf-friends, specifically Frodo and Aragorn.