The reasons for Galadriel originally being named in the Telerin language are complex. In the published Silmarillion, Galadriel's spouse Celeborn was a Sinda of Doriath, whom she met after she had arrived in Middle-earth. However, it seems to have been Tolkien's intention to change this, and make Celeborn originally a Teler of Aman, who gave her this name in his own tongue, and later accompanied her into exile.
This alternative account was never incorporated into the main Silmarillion tradition, but it's still possible to account for Galadriel's originally Telerin name. Her mother Eärwen was one of the Teleri, and because of this, it's said that the Telerin language was used in Finarfin's house. It's not hard, then, to imagine Alatáriel acquiring her name from some member of her father's household, and then carrying it across the Great Sea with her as the origin of Galadriel.
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