Avraham first has a child with Hagar -- the same Hagar who was giften as a maid servant by Par'oh after the events in Egypt. The only discussion of her name that I saw was in the Chizkuni (on Bereishit 16:1) who says that Par'oh made her a reward to Sarah (with the word for reward as אגרך, therefore, her name was Hagar).
This suggests that she had no name beforehand, or that she had another name (was Keturah her birth name?), as she is given this name only when she is handed over by her father. Are there any sources which discuss her name more thoroughly?
It seems to me that she was the one with whom Avraham "settled" -- not permanently residing (in the same way that he Hagada mentions that Yaakov went to Egypt lagur, to settle, but not to stay -- שֶׁלֹא יָרַד יַעֲקֹב אָבִינוּ לְהִשְׁתַּקֵּעַ בְּמִצְרַיִם אֶלָּא לָגוּר שָׁם) and since she is named at some point after her birth, it makes as much sense to give her this name to show she isn't really Avraham's "real" wife as it does to name her to show that she is a gift to Sarah.