Avarham Avinu was originally Avram - and Hashem added the letter Hai to his name and changed it to Avraham. Why from all letters was the letter Hai added?
The name אברם numerically equals 243. Originally Avram was master of 243 of his 248 body parts: all except his two eyes, two ears, and male organ. [These are usually exposed to improper stimuli, even against a person's will.] However, with the Hei (numerical value of 5) added to his name, Hashem granted him control even over these [so that he no longer could see or hear anything that he shouldn't].
(Nedarim 32b and Tosafos there)
Chasam Sofer - Parshas Lech Lecha - Hishaleich says that as Sarai became Sarah - the Yud switched into a Hei. The remaining 5 went to Avraham.
I heard in a shiur from R' Moshe Wolfson that Avraham was given his new name in the context of the promise to have children and become a nation. The letter ה is the אות ההולדה, the letter of birth, as it is the feminine letter (the letter that turns a word into a grammatically feminine word). Avraham was being given the ability to father a nation, symbolized by the addition of the ה to his name.
R. Yitzchak Hutner (in Pachad Yitzchak to Sukkos) writes that the reason a "hei" was added was because Avraham was becoming a new creation, so to speak - being fashioned from new. As the Midrash writes, (Midrash Rabba 12:2) when God created the world, He did so with the letter "hei". Whatever that may mean, the letter "hei" is clearly symbolic (and maybe even metaphysically related) to recreation.
R. Hutner uses this idea, interestingly, to provide a source for the rule of the Rabbis that a convert who converts is like a newborn baby (regarding their previous familial relationship) and thus a new creation.
It says expressly in Yerushalmi Sanhedrin 13a / 2:6:
יו"ד שנטל הקב"ה מאמנו שרה ניתן חציו על שרה וחציו על אברהם
The 'Yud' that Hashem took from Sarah (i.e. her previous name was Sarai with a yud at the end) was given half to Sarah and half to Avraham.
In other words, the Yud which has a numerical value of 10 was removed and split in two to becomes two Hei's (i.e. two sets of 5) of which one hei went to the end of Sarai's name to become Sarah and the other hei went to Avram to become Avraham.