The Living, The Alive, The Everlasting.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Hayy (in Arabic ٱلْحَىُّ), the One who is undying and will be forever. He is perfect and not subject to death or sickness. He is eternally whole and absolute, whom all life arises from.
From the three-letter root ha-ya-ya (ح ي ي), which has the following classical Arabic connotations: to live, to be living, to become apparent, distinct, to animate, to revive, to vitalize, to call into being, to be whole, sound, to call, summon, invite, hasten.
As we discussed in the previous two names, Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Muhyi meaning, the giver of life, and Al-Mumeet, the bringer of death. This name, Al-Hayy describes the quality of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as being ever-living. There is no beginning and end to His existence. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ just is. There are many layers to the name Al-Hayy which we will explore in this section. (1) It Has already been mentioned and discussed extensively throughout the previous names; it's Allah's essence or quality of being everlasting. (2) The second layer is the perfection of His life. (3) The third quality is being the source of life.
The word Hayy comes from hayaat which means life. One of the criteria of life as we know it is to have perception. Imam al-Ghazali says, "The lowest level of perception involves the one perceiving being conscious of itself." Plants are said to be alive since they detect and orient themselves toward the sunlight. Living things are also defined by their ability to respond purposefully to what they perceive. But all living things are limited in their scope of activity and perception. We operate within narrow and defined limits by the creator. And Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ in His infinite wisdom has given life to a diverse range of creations which vary in these two qualities of perception and activity. Shaykh Tosun Bayrak mentions how a plant is superior to the soil or rock since it's given the faculties to eat, drink and propagate. Life in animals is more developed with hearing, seeing, and moving. And so naturally, animals learn to dominate over vegetation. Human life is said to be the highest form of Allah's creation. We can store what we perceive and use our intellect, reasoning, and free will to respond with this information. We can act consciously based on our knowledge, a quality that seems to be lacking in animals who only respond to their desires. People differ in their "aliveness," which is often a function of their knowledge and actions.
Now, when the word hayy is used as a name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ it shows how He is aware of all things. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the most perceptive. He is As-Sami (The All-Hearing) and Al-Baseer (The All-Seeing). There is nothing that escapes His knowledge as He is aware of all activity. In this sense, it can be said He is the absolutely living. This transcends all creation, not just in the sense of being everlasting, but in terms of what it means to be perfect in living. His life is perfection; it is not touched by sickness, weakness, or any deficiency. Absence does not precede Him, nor does death follow after Him. In Surah Baqarah verse 255 (Ayatul Kursi), Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ says:
Allahu laaa ilaaha illaa Huwal Haiyul Qaiyoom; laa taakhuzuhoo sinatunw wa laa nawm; lahoo maa fissamaawaati wa maa fil ard; man zal lazee yashfa’u indahooo illaa bi-iznih; ya’lamu maa baina aydeehim wa maa khalfahum wa laa yuheetoona bishai’im min ‘ilmihee illaa bimaa shaaa’; wasi’a Kursiyyuhus samaawaati wal arda wa laa Ya’ooduhoo hifzuhumaa; wa Huwal Aliyyul ‘Azeem
"Allah – there is no deity except Him, the Ever-Living, the Sustainer of [all] existence. 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 Kursi extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great." — (Qur'an 2:255)
This is just one of five verses in the Qur'an where the name Al-Hayy is mentioned.
Allaahu laaa ilaaha illaa Huwal Haiyul Qaiyoom
"Allah – there is no deity except Him, the Ever-Living, the Sustainer of existence." — (Qur'an 3:2)
Wa ‘anatil wujoohu lil Haiiyil Qaiyoomi wa qad khaaba man hamala zulmaa
"And [all] faces will be humbled before the Ever-Living, the Sustainer of existence. And he will have failed who carries injustice." — (Qur'an 20:111)
Wa tawakkal ‘alal Haiyil lazee laa yamootu wa sabbih bihamdih; wa kafaa bihee bizunoobi ‘ibaadihee khabeeraa
"And rely upon the Ever-Living who does not die, and exalt [Allah] with His praise. And sufficient is He to be, with the sins of His servants, Acquainted –" — (Qur'an 25:58)
Huwal Hayyu laaa ilaaha illaa Huwa fad’oohu mukh liseena lahud-deen; alhamdu lillaahi Rabbil ‘aalameen
"He is the Ever-Living; there is no deity except Him, so call upon Him, [being] sincere to Him in religion. [All] praise is [due] to Allah, Lord of the worlds." — (Qur'an 40:65)
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ essence is living. He is the one responsible for all creation. There would be no life without His willingness to allow it. So we recognize Him as being the only source we should turn to and rely upon for our sustenance. In fact, whenever the Prophet ﷺ was stressed, He used to call upon the two names, Ya Hayyu, Ya Qayyum (i.e., the subsisting, the self-sustaining).
Anas bin Malik said:"Whenever a matter would distress him, the Prophet (ﷺ) would say: 'O Living, O Self-Sustaining Sustainer! In Your Mercy do I seek relief (Ya hayyu ya Qayyum, bi-rahmatika astaghith).’” And with this chain, that he said: "The Messenger of Allah (ﷺ) said: 'Be constant with: "O Possessor of Majesty and Honor (Ya Dhal-Jalali wal-Ikram).'" 
Another hadith mentions a second du'a which includes reciting these two names. Narrated Anas ibn Malik: I was sitting with the Messenger of Allah ﷺ and a man was offering prayer. He then made supplication: O Allah, I ask Thee by virtue of the fact that praise is due to Thee, there is no deity but Thou, Who showest favour and beneficence, the Originator of the Heavens and the earth, O Lord of Majesty and Splendour, O Living One (ya hayyu), O Eternal One (ya-qayyum). The Prophet ﷺ then said: He has supplicated Allah using His Greatest Name, when supplicated by this name, He answers, and when asked by this name He gives.
It is conferred from the last hadith that the greatest names of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ may be Al-Hayy and Al-Qayyum. This is consistent with another narration by Al-Qasim, who said: The Greatest Name of Allah, if He is called by which He will respond, is in three Surah: Al-Baqarah, Al-Imran and Ta-Ha. (Hasan) Another chain for something similar from Al-Qasim, from Abu Umamah, from the Prophet (ﷺ).  There is another view that Allah may be the greatest name of all, and the case for that is because it is mentioned 2698 times throughout the Qur'an, way more than any other name. Regardless, it is not a worthwhile pursuit to view the names of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as being hierarchical and trying to determine their rank. Instead, we should follow the advice given in the Qur'an, "And to Allah belong the best names, so invoke Him by them." (7:180) Similarly, we should not try to rank Allah's creation and feel superior. Turn this awareness into gratitude and praise Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ for the gifts given. The emphasis of the hadith could be that invoking these names in our du'a holds extra weight or reward for the believer. Especially when said in a time of distress when you need Him the most.
The continuation of Islam after the death of Prophet Muhammad ﷺ:
There is an important message in the Qur'an and hadith. You can imagine after the death of Prophet Muhammad ﷺ, some people whose faith hadn't been fully nourished yet felt disconnected. They listened to Prophet Muhamamd for his leadership and guidance, and when he returned to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ they felt lost. Surah al-Imran ayah 144 reminds all of us, "Muhammad is not but a messenger. [Other] messengers have passed on before him. So if he was to die or be killed, would you turn back on your heels [to unbelief]?" Abu Bakr (the first caliphate to succeed Prophet Muhammad ﷺ) came to His funeral prayer
Narrated Aisha (ra): Abu Bakr came riding his horse from his dwelling place in As-Sunh. He got down from it, entered the Mosque and did not speak with anybody till he came to me and went direct to the Prophet, who was covered with a marked blanket. Abu Bakr uncovered his face. He knelt down and kissed him and then started weeping and said, "My father and my mother be sacrificed for you, O Allah's Prophet! Allah will not combine two deaths on you. You have died the death which was written for you." Narrated Abu Salama from Ibn Abbas: Abu Bakr came out and Umar, was addressing the people, and Abu Bakr told him to sit down but Umar refused. Abu Bakr again told him to sit down but Umar again refused. Then Abu Bakr recited the Tashah-hud (i.e. none has the right to be worshipped but Allah and Muhammad is Allah's Messenger ﷺ and the people attended to Abu Bakr and left Umar. Abu Bakr said, "Amma ba'du, whoever amongst you worshipped Muhammad, then Muhammad is dead, but whoever worshipped Allah, Allah is alive (Al-hayy) and will never die. [here ayah 3:144 was recited] (The narrator added, "By Allah, it was as if the people never knew that Allah had revealed this verse before till Abu Bakr recited it and then whoever heard it, started reciting it.") 
This can be a very instructive lesson for parents with children. In the same sense, they are guardians/leaders over their children, teaching them Islam just as the Prophet ﷺ taught the sahabah. Sometimes children don't understand or become disconnected from the fact that worship is for Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ and not for seeking parental approval - or in some cases, avoiding parental punishment. Once the parent passes away or if the children grow up, get married and move away, they should remember that it is Al-Hayy, the ever-living, that they're seeking approval from.
Many go through life at the whims of their impulses and desires. The one who is truly alive is the one who tames himself. His life belongs to himself, and he is not a slave to whatever the controlling force of the time may be. The truly living are the ones who are in constant remembrance of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. For them will be granted eternal paradise in the hereafter.