The Elamites encouraged the governor of Babylon to lead a revolt
against then-King Assurbanipal of Assyria that resulted in the
destrction of Elam. Name that governor.

lisanne --

The answer to your question is Shamash-shum-ukin, who was the brother
of King Assurbanipal of Assyria (also spelled Ashurbanipal).

Although the historical accounts of this period in the 7th century,
B.C. differ somewhat in the details, the essence of the story is
confirmed in a variety of source materials, cited below.

Esarhaddon, the father of Shamash-shum-ukin and Assurbanipal,
bequeathed Babylonia to Shamash-shum-ukin and Assyria to Assurbanipal.
  Shamash-shum-ukin's relationship with his brother deteriorated, and
Shamash-shum-ukin began a secret alliance with the Elamites, the
Arameans and others, directed against his brother Assurbanipal.

Earlier, Elam had been defeated by Assurbanipal's forces following an
Elamite incursion, so Elam was naturally eager for the alliance headed
by Assurbanipal's resentful brother to revolt against him.

This revolt ended badly for Shamas-shum-ukin, whose army was defeated
in 648 B.C. and who died in his burning palace in Babylon.

In the following year, the Elamites were punished with their final
destruction by Assurbanipal, whose army destroyed Susa, Elam's
capital, and Elam became an Assyrian province.

The following sources were used to put together the above summary:

The Assyrian Empire 
Ashurbanipal (668-627) and Shamash-shum-ukin (668-648) (about 3/4 of
the way down the page)
http://www.angelfire.com/nt/Gilgamesh/assyrian.html

The Immanuel Velkovsky Archive: Seti Becomes an Ally of Assurbanipal
http://www.varchive.org/tac/seti.htm


Additional Links:

Encyclopedia.com: Assurbanipal
 http://www.encyclopedia.com/html/A/Assurban.asp

Slider.com: Elam
http://www.slider.com/enc/17000/Elam.htm


Google Search Terms Used:

assurbanipal elam
://www.google.com/search?sourceid=navclient&q=assurbanipal+elam

assurbanipal shamash-shum-ukin
://www.google.com/search?sourceid=navclient&q=assurbanipal+Shamash%2Dshum%2Dukin

This was a fascinating question and a learning experience for me.  If
any of the above needs clarification, I would be happy to provide it.

markj-ga

I was not sure if shamashshumukin was the correct answer, but with
this clarification I beleive it is. Thank you very much!