Gandalf says of himself:
"Many are my names in many countries. Mithrandir among the elves, Tharkûn to the dwarves; Olórin I was in my youth in the West that is forgotten, in the South Incánus, in the North Gandalf, to the East I go not."
The Two Towers
Further, this answer includes the following quotes:
The last-comer was named among the Elves Mithrandir, the Grey Pilgrim, for he dwelt in no place, and gathered to himself neither wealth nor followers, but ever went to and fro in the Westlands from Gondor to Angmar, and from Lindon to Lórien, befriending all folk in times of need...
Mostly he journeyed unwearingly on foot, leaning on a staff; and so he was called among Men of the North Gandalf, “the Elf of the Wand”. For they deemed him (though in error, as has been said) to be of Elven-kind, since he would at times works wonders among them, loving especially the beauty of fire; and yet such marvels he wrought mostly for mirth and delight, and desired not that any should hold him in awe or take his counsels out of fear.
Unfinished Tales Part 4 Chapter II: "The Istari"
So Mithrandir is Sindarin for "Grey Pilgrim", which is why the Elves know him by it, and both Men and Hobbits call him Gandalf, meaning "Elf of the staff", and the Dwarves have another name for him.
However, I noticed that Gondorians refer to him as Mithrandir at least three times in the Extended Edition of the movie - by Denethor at
00:46:55, by Faramir at
1:13:50, and by a Gondorian guard at
Is there a reason given in the source material that explains why the Gondorians often refer to him as Mithrandir, where the Rohirrim - the other kingdom of Men - do not?