The Etymology of Avraham

The pesukim state (Bereishit 17:3-5):

ג וַיִּפֹּל אַבְרָם, עַל-פָּנָיו; וַיְדַבֵּר אִתּוֹ אֱלֹקִים, לֵאמֹר.	3 And Abram fell on his face; and God talked with him, saying:
ד אֲנִי, הִנֵּה בְרִיתִי אִתָּךְ; וְהָיִיתָ, לְאַב הֲמוֹן גּוֹיִם.	4 'As for Me, behold, My covenant is with thee, and thou shalt be the father of a multitude of nations.
ה וְלֹא-יִקָּרֵא עוֹד אֶת-שִׁמְךָ, אַבְרָם; וְהָיָה שִׁמְךָ אַבְרָהָם, כִּי אַב-הֲמוֹן גּוֹיִם נְתַתִּיךָ.	5 Neither shall thy name any more be called Abram, but thy name shall be Abraham; for the father of a multitude of nations have I made thee.

His name used to be Avram, and now Avraham. It sounds similar, and is the expansion of the name, but it connotes this additional message. The message is by sound similarity, or at least seems to be. Av is there, from father, and Ham is there, which starts of Hamon. The resh is still there, somewhat problematically, but not really so, since Arvaham is a fine, fuller version of the earlier name.

Rashi states something similar:

the father of a multitude of nations אַב הֲמוֹן is an acrostic of his name [i.e., - אב ר הם]. (Gen. Rabbah 46:7). The “resh” that was in it [his name] originally, denoting that he was the father only of Aram, which was his native place, whereas now [he became] the father of the whole world (Ber. 13a): nevertheless the “resh” that was there originally was not moved from its place. For even the “yud” in Sarai’s name complained to the Shechinah until it was added to Joshua, as it is said: (Num. 13:16):“and Moses called Hosea [הוֹשֵׁעַ] the son of Nun, Joshua [יְהוֹשֻׁעַ].” - [from Gen. Rabbah 47:1]

While he mixes in midrash in there, he is saying peshat (or midrasho-pshat) here in the general sense.

Shadal gives an interesting etymology:

ריהאם בלשון ערבית כמו המון בעברית ענינו ריבוי ומספר גדול

He notes that Riham, that is RHM, in Arabic, means Hamon. Thus Av-Riham is "father of a multitude." He is not saying that Hashem gave Araham this new name in Arabic, but just that this is a Semitic cognate and thus the root was available in Hebrew, even though the simpler Hebrew word equivalent is given in the explanation.

Ibn Ezra notes the resh as well, and answers that it is really "abir hamon goyim." He would somehow work out that av is used in the verse, perhaps by saying that av can mean abir in some sense of mastery, or some other explanation like this. He adds that Hashem came to add a letter, not replace it.