Elfstone rejoices! Finally something besides Lembas to sink his teeth into again!
Well having recently read The History Of Galadriel And Celeborn in UT, and not to mention that I’m a die-hard Galadriel fan, I feel impelled to throw in my two cents.
For starters, in Valinor concerning the customs of name giving among the Eldar, Elves were usually given two names (essi), of which the first was given by the father (generally recalling the father’s own name in some shape or form), and the second name was given later by the mother (sometimes much later, but sometimes very soon after the birth). The mother names were often very significant because the mothers of the Eldar possessed great insight into the characters and abilities of their children, and many had the gift of prophetic foresight.
In addition, any of the Eldar might also acquire an after-name (epesse), not necessarily given by their own kin, a nickname generally given as a title of honor and admiration. The epesse might become in some cases the name generally used and recognized in later song and history. For example Ereinion, who was always known, and referred to by his epesse Gil-galad.
My beloved Lady of Light’s father name was Artanis, and her mother name was Nerwen. Alatariel was an epesse (nickname) given to Galadriel by Celeborn (in the later version of their history) in Aman, which she choose to use in M.E. This name in Sindarin translates to Galadriel.
I believe that Allyssa, and Arcormacolindova have already done an excellent job of explaining the reason/s for the difference between Galadriel and Galadhriel. Oops, which was the specific question at hand.
"I would have the Ring-bearer bring the crown to me, and let Mithrandir set it upon my head, if he will; for he has been the mover of all that has been accomplished, and this is his victory." Elessar