The name 'Azra'il is used for the Angel of Death in a number of hadith, such as in Bihar al-Anwar, but not in the Qur'an.
A du'a transmitted from Imam al-Sadiq in which one prays for blessings on the angels includes praying for blessings on 'Azrail, the Angel of Death (allahumma salli 'ala Azra'il, malak al-mawt).
Interestingly, there is a variant of this du'a which calls him "malak al-rahmah" - the Angel of Mercy, although still describing him as taking people's souls. Perhaps this is because, for the believer, it is said that he will be kindly.
In any case, he is doing his job tirelessly and deserves blessings! :)