Who gave the name "ALLAH " or did he keep it to himself.

I do not know if the questioner has ever read the Quran.  If you read these verses, you should read -

 1. All praise is due to Allah, the Lord of the worlds.  (Surat al-Fatihah, Ayat 1)

 2. Say: Allah is One and Unique. (Surah Al-Ikhlas, verse 1)

 3. Allah, there is no true god but He, the Sustainer of all life.  (Surah: Ale Imran, verse 2)

 4.  And for Allah belong the best names.  So call him by those names.  And shun those who pervert His name.  They will soon be recompensed for what they used to do.  (Surah: Al-A\"raf, verse 160)

 There are many more such verses in which Allah Himself has given us His own name.  Not only this name "Allah" but Allah has 99 other names which Allah has informed us in different verses of different Surahs.

 Since the Qur\"an is sent down by Allah and there is no one to change it or no one can add or remove any new words to it.  So it is not acceptable for someone else to give the name of Allah.

 I do not know the purpose of such questions.

 "Does not man see that We have created him from a drop of semen? Then he became an open disputant. He narrates a strange thing concerning me, while he forgets his own creation."