What sort of name is קליר? There have been several suggestions. The first, quoted in the Jewish Encyclopedia, is in the name of the 'Arukh (11th century), who derives the name from the Greek kollura κολλuρα "a small cake," claiming that he obtained his name from eating a cake inscribed with Biblical verses, which was given to him to eat (as a child?) as a sort of charm to obtain wisdom: יש מקום שקורין לעוגה קליר על כן נקרא רבי אליעזר הקליר שאכל עוגה שהיה כתוב בה קמיעא ונתפקח
However, this suggestion is difficult because in his acrostics Kalir typically spells out אלעזר ברבי קליר ** indicating that קליר is to be understood by us as his father's name. As an aside, in a few of his acrostics קליר is spelled קיליר, indicating that is Kilir is probably more correctly how this name was pronounced.
Another derivation is suggested by Paul Kahle (The Cairo Geniza pg. 20 n.3):
[I]n old Geniza fragments the name is given in the acrostics as Kilirr (קילירר) with double r, and this shows clearly that the name goes back to Cyrillus, with metathesis of r and l.
I'm not sure how clearly it shows it relative to how clearly it is claimed, but we're getting somewhere. Least compelling in my view is that it is derived from the name of the Italian city Cagliari.
*Poetan פיטן; to use the charming orthography preferred by English writers of the 19th century
** Eleazar son of Rabbi Qalir