The Prophet ﷺ said: “The most beloved of your names to Allah are ʿAbd Allah and ʿAbd al-Rahman” (Muslim 2132), and scholars add to them all names formed on a name of Allah. The name al-Hayy occurs 5 times in the Quran; its page in the 99 Names section gives its meaning and verses.
ʿAbd al-Hayy joins ʿabd with al-Hayy, from the root h-y-y of life: the One whose life is perfect, without beginning or end, who does not die. The Throne Verse opens with this name: “Allah, there is no god but He, the Ever-Living, the Sustainer of all” (2:255).
Allāh - there is no deity except Him, the Ever-Living, the Self-Sustaining. Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them, and they encompass not a thing of His knowledge except for what He wills. His Kursī extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great.