Abdul Alhazred was a pseudonym adopted by Lovecraft after reading 1001 Arabian Nights in his early childhood. The name may have been invented by Lovecraft himself or the Phillips' family lawyer Albert Baker.
Abdul (Abd al) is a common Arabic name component (meaning "servant of the") but never a name by itself. Alhazred may allude to "Hazard", a reference to the book's destructive and dangerous nature, or to Lovecraft's ancestors by that name. It might also have been a play on "all-has-read", since Lovecraft was an avid reader in youth.
Another possibility, raised in an essay by the Swedish fantasy writer and editor Rickard Berghorn, is that the name Alhazred was influenced by references to two historical authors:
Alhazen ben Josef, who translated Ptolemy into Arabic; and Abu 'Ali al-Hasan ibn al-Haytham, who wrote about optics, mathematics and physics (both authors were known in the Latin West as "Alhazen", although the latter was substantially more popular than the former). Ibn al-Haytham is said to have pretended to be mad to escape the wrath of a ruler.
Abdul Alhazred is not a real Arabic name, and seems to contain the Arabic definite article morpheme al- twice in a row (anomalous in terms of Arabic grammar). The more proper Arabic form might be Abd-al-Hazred or Abdul Hazred. In Arabic translations, his name has appeared as Abdullah Alḥa ẓred (عبدالله الحظرد): Arabic ḥaẓaraحظر = "he fenced in", "he prohibited". Hazred could come from the Arabic word "Hazrat" meaning Great Lord with a twist that makes it sound like "red" and "hazard" both indicative of danger. It is also thought by some to be a corruption of sorts on the phrase "All has read", to imply he has read much, and has immense amounts of knowledge. However Abdul is a common Arabic prefix meaning "Servant" and "Al" is Arabic for "the", and if "hazra" means "he prohibited", "he fenced in" or "Great Lord", then the name would mean "Servant of the Prohibited", "Servant of the Fenced in", or "Servant of the Great Lord" which would make sense considering his role, even if it is not a proper Arabic name.
An explanation that is more in sync with Arabic usage and existing Sufi tradition is that it is a corruption of "Abd-al-Hazra[h]" عبدالحضرة, where "haẓrat" is the Persian and Ottoman Turkish form of the Arabic word "Haḍra[t]" | Hadrat حضرة meaning "presence" used by some speakers as an honorific title before the names of prophets, saints, and also as a mnemonic for the name of Allah, as well as a common honorific title for ordinary people. The final taa marbuta is customarily variably turned into "t" or omitted in spoken Arabic in various varieties. "Haḍra" is also the name of the Sufi Dhikr.
The phrase "mad Arab", sometimes with both words capitalized in Lovecraft's stories, is used so commonly before Alhazred's name that it almost constitutes a title. A reference to the "Mad Arab" in Cthulhu Mythos fiction is invariably a synonym for Abdul Alhazred. Later writers sometimes preface Alhazred with words such as "monk" (such as in the "Who will be Eaten First?" by Howard Hallis) or "scholar".