I have long been intrigued by the given name of Azerbaijan's first president, Abulfaz. What is its origin, or etymological meaning? Would it be Arabic "Abu -something"?

One of the sons of Ali ibn Abi Talib, the fourth rightly-guided caliph or first imam apparently has the title Abul Fazl, which is there transliterated Abu Fadhl, but z is equally possible for the letter transliterated dh, and there translated Father of Virtue, although I would have said Father of Grace.
It could be that this is an Azerbaijani reduction of that. I don't have a dictionary handy and don't know that there isn't an Arabic name/word faz but that doesn't mean that there isn't.
Digression on abu- names. In this case the name is a title. In most cases an Arabic name in Abu is a kunya, a nickname, given after the birth of the first-born son, meaning "father of x." Abu Talib, the father of Ali mentioned above, presumably had a son Talib (=seeker, student). The feminine equivalent is Umm, mother.
Occasionally someone known by such a kunya will become famous is his or her own right and the kunya will come to be used as a given name. An Abu Bakr probably is named after the caliph Abu Bakr, an Umm Kulthoum is probably named after Umm Kulthoum, a wife of the prophet Muhammad; they probably don't have sons called Bakr or Kulthoum.

Another note. Thorn Tree wouldn't let me put apostrophes in the link or I would have written that Abu'l Fazl. The 'l is the definite article. Abu = father.
Now we'll wait for the right answer from hlatif.

Checked a dictionary and I don't see any likely faz. And I stand by "grace" rather than "virtue" for fadhl.

How is the real letter that is transcribed variously as z or dh (or sometimes just d i think) pronounced? somewhere between a european d and z?
And what's its name?

What VinnyD wrote is correct. The only correction is that Um Kulthum was actually the daughter of the prophet. The letter in fadhl is Dhad and in Arabic it is proununced closest to the D in Duh. There are two other Arabic letters that are changed to Z in other languages:
TH as in "the"
THA which is the same as above but thicker.
I hope this helps
Hussein

Um Kulthum was actually the daughter of the prophet
Last night I started worrying that that might be the case. I figured no wife of the prophet could have had a son Kulthum. (He had a son Ibrahim who predeceased him.) I didn't think he married any widows after Khadijah and I didn't think any of his widows remarried. So the name was hard to explain unless I had it wrong.
Thanks.
The dh letter in fadhl is the same as the one in Ramadhan/Ramadan. I think Turkish regulary makes it a z, as in Ramazan. Azeri probably does too.

Actually, most of his wives were either widowed or divorced. One of them was the very important Um Salama who was widowed with several young kids. He took them into his household. Um Salama was known as being a very smart and intellectual woman and her daughter was one of the most important women scholars of Medina of her time.
Hussein
