Discussion of Hagar's name

Question

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.

Answer 1

Malbim says that it indicates that she did not convert:

ושמה הגר ר"ל שלא נתגיירה כי לא נשתנה שמה שבעת הגרות היו משנים את השם

Shadal cites an explanation that the name Hagar is a post facto application based on the events that occurred — in Arabic it means to run away. But Abraham and Sarah never called her by any name — they would refer to her simply as "my maidservant" or "your maidservant":

ושמה הגר יש אומרים שלא היה לה שם זה מתחילתה אלא שנקראה כן אח"כ על שם המאורע כי הגר בערבי ענינו ברח והלך אל ארץ אחרת (קרוב לשורש גור בעברי) ומזה היג'רה אל נבי היא

Egira

שתחילתה משעה שברח מוחמד מן

Mecca

בשנת 622 למספרם ובאמת אברהם ושרה קוראים אותה תמיד שפחתי שפחתך ולא מצאנו בשום מקום שיזכירו שם הגר אע"פ שאין זו ראיה