Is there a name for The God Who Hears? I was wondering if there is a name meaning God Who Hears me.

El Roi speaks of The God Who Sees me.  From Genesis 16:13 (ESV):


13 So she called the name of the LORD [El-Roi] who spoke to her, "You are a God of seeing," for she said, "Truly here I have seen him who looks after me."


Googling God Who Hears turned up one site that (incorrectly?) said Jehovah-Shammah (usually translated The Lord Who Is Present or The Lord Is There) meant God Who Hears me.  Their given verse was Psalm 17:6 (ESV):


6 I call upon you, for you will answer me, O God; incline your ear to me; hear [shama`] my words.


I don't know Hebrew at all, but checking Strong's online out of curiosity, [shama`] and [shammah] seem (to me) to be unrelated words.

Does anyone know of a name that meant God Who Hears me?
You are close: 'El Shama is what you are looking for.  Shama means "to hear" (Strong's) and is found in Psalm 17:6, as you mention.  Furthermore, the name Ishmael is derived from these two words:


  And the angel of the Lord said to her, “Behold, you are pregnant and shall bear a son.
  You shall call his name Ishmael, because the Lord has listened to your affliction. (Genesis 16:11 (ESV))


The ESV note specifically says, "Ishmael means God hears," and so does Strong's Concordance.

There's also the name "Elishama," meaning "God has heard," which belongs to several Israelites listed in the genealogies of the OT (e.g., Numbers 1:10 and 1 Chronicles 2:41).

The "Shammah" rendering you saw may be due to the pronunciation; Strong's gives the phonetic spelling of Shama as "shaw-mah."