Lovecraft has taken a lot of criticism over the years for the name he gave to the author of the Necronomicon. Sure, it does not follow actual Arab naming convention (the -el suffix and the Al- prefix are redundant for example), but a quick look at Latinized names of other Muslim figures shows a similar disregard. I have heard a number of origin stories going all the way back to Lovecraft's childhood, and while they are interesting ("all-has-read" being particularly clever), I feel he probably was inspired by the name Shahrazad from his beloved copy of Arabian Nights Entertainments.
In my research just now I stumbled across this fascinating fellow who not only shares the first five letters with Alhazred's name, but authored books while feigning madness! Could Lovecraft have known of him? I don't know, but the coincidence was too good not to share.