The phrase "everlasting life" appears once in the OT, in Daniel 12:2. Jesus' usage in the gospels is brief. What did the ruler hear that made him rush to Jesus to ask for it? Are there any clues I missed that give us a fuller picture? Are there Christian writings that have addressed this point?
The Pharisees were the dominant religious sect of Judaism in Jesus's time. They taught strict adherence to the Law and all of the other laws added to it. They also taught the resurrection of the dead. In their contemporary writings, the Pharisees talk about eternal hell and they used very specific terms we know mean eternal hell because they explain it at length. They also believed in eternal life as an alternative if one followed the laws to a T.
The rich young ruler said he'd kept the Commandments since he was a boy, so it's possible he held Phrisetical beliefs. He would have also heard teachings about eternal life from the Pharisees.
It's important to note that Jesus did not use the same terms to describe hell that the Pharisees used. Jesus used words that normally mean age-bound correction. But due to doctrinal beliefs shifting in the 5th Century A.D., the Latins began translating His words as eternal judgment and eternal punishment. Due to the difference in terms that Jesus used, it's possible the rich young ruler was drawn to Jesus' more loving teachings of God rather than the harsh Pharisetival teachings he'd grown up seriously adhering to. The Greek Church Fathers taught that hell was temporary and everyone there would eventually be purified and go to heaven. They were the first Church Fathers and spoke the New Testament's Koine Greek as their native language and therefore understood it perfectly well.
Also, in the Old Testament, eternal life doesn't appear, just as eternal torment doesn't appear either, not when translated properly. The Ancient Hebrews of Abraham's day were nomads and their Paleo-Hebrew language (Pheonecian) was built around the way they thought and lived--with nomadic terms.
"Olam/Owlam" is their word we translate to mean eternal. However, "olam" doesn't mean eternal. It literally means "over the horizon," which is their way of saying "future," because they didn't know what was over the horizon until they got there. They had another saying that meant the distant future (I forget the exact phrase), which literally means "over the horizon and back." In their culture, they didn't speculate on things they didn't know for certain. So they simply described the afterlife as the unseen or as the future or distant future.
Life was described the same way, as "future" life or "distant future" life. Our translators simply translated their mainstream doctrinal and cultural beliefs into the texts, which is nearly impossible not to do if you understand how translation must be done and how much has to be guessed at due to cultural differences and unknowns such as their exact beliefs back then. Translation isn't nearly as simple as "Casa" means "home" in Spanish. It's extremely difficult and pretty much impossible to get a 100% accurate translation.
Despite what I just said about the word eternal, that doesn't mean eternal life isn't stated in the Bible. When the Koine Greek word aion/aionios/aionion (age or age-bound) is paired with something we know to be eternal (like God), then we translate aionios as "eternal." The problem, though, is that we don't know for certain if judgment or punishment is eternal, yet we translate aionios to mean "eternal." Since the Church Fathers use the term aionios punishment/judgment just like Jesus but in the same sentence say that the person being judged/punished is purified by it and then goes to heaven, this likely proves that the term doesn't actually mean "eternal" when paired with judgment/punishment.
Furthermore, the Koine Greek words Jesus & Paul used for punishment and judgment are corrective terms rather than vengeful terms. Corrective implies an end purpose of purification rather than annihilation or eternal torment.
In other words, the word used to describe eternal is a very controversial term, but when used in relation to life, we're pretty sure it means eternal life, but in the Old Testament, we don't know for certain that the idea of eternal life, as we conceptualize it, is ever used. We do know that the general beliefs and ideas from the Old Testament are being conveyed in the New Testament by Jesus and the other writers, though, because they quote the Old Testament so frequently.
I hope that wasn't too in-depth or too widely focused to answer your question to your liking.
The young man might have read the Book of Parables of Enoch (1 Enoch 37–71) (also called the Similitudes of Enoch), which in the Third Parable (ch. 58-71) deals with escatology. Ch. 58 mentions eternal life explicitely:
And the righteous shall be in the light of the sun. And the elect in the light of eternal life: The days of their life shall be unending, And the days of the holy without number. (1 Enoch 58:3)
In the present state of scholarly research on the Book of Parables, the position bearing the greatest weight is that the book was written in Galilee towards the end of the kingdom of Herod the Great (37 BC - 4 BC) or shortly after his death .
 Darrell L. Bock and James H. Charlesworth (ed.), Parables of Enoch: A Paradigm Shift, Bloomsbury, 2013. https://books.google.com/books?id=PW3roOm3LG0C
The phrase "everlasting life" appears once in the OT, in Daniel 12:2.
In the OT,King David suggest that some people decay in the grave while others are redeemed from the grave,if somebody is taken by God from death means one thing,that person will get to live for eternity with his God.The tree of life can also give eternal life(if The Lord allow you to eat from it).The OT also talk about how God can't be bribed to give a person a chance to have his soul redeemed from death(and live on forever)
Like sheep they are destined for Sheol. Death will be their shepherd. The upright will rule them in the morning, and their form will decay in Sheol, far from their lofty abode. But God will redeem my life from Sheol, for He will surely take me to Himself. (Psalm 49).
And the LORD God said, "The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever."Genesis 3:22
No man can possibly redeem his brother or pay his ransom to God. For the redemption of his soul is costly,and never can payment suffice,that he should live on forever and not see decay.Psalm 49:9
Jesus' usage in the gospels is brief
It isn't brief,the whole gospel is based on the fact that whoever believe in Jesus shall have eternal life.
Jesus said to her, “I am the resurrection and the life. He who believes in Me will live, even though he dies.And everyone who lives and believes in Me will never die. Do you believe this?” John 11:26
"Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up, that everyone who believes in Him may have eternal life.For God so loved the world that He gave His one and only Son, that everyone who believes in Him shall not perish but have eternal life."John 3:15
Truly, truly, I tell you, if anyone keeps My word, he will never see death.”John 8:51 And He has given Him authority to execute judgment, because He is the Son of Man. Do not be amazed at this, for the hour is coming when all who are in their graves will hear His voice and come out—those who have done good to the resurrection of life, and those who have done evil to the resurrection of judgment. (John 5:28)
Here are verses about the tree of life we saw early in genesis.It is offered as a gift by the Lord in revelation.
Blessed are those who wash their robes, that they may have the right to the tree of life and may go through the gates into the city.Revelation 22:14
What did the ruler hear that made him rush to Jesus to ask for it? Are there any clues I missed that give us a fuller picture?
He might have been convicted in his heart(just guessing).Despite his good deeds,he still couldn't shake this feeling of not doing enough to inherit eternal life.God gave him an answer but he refused.
Side note:Eternal torment in the OT.
The cool thing about God is how He never changes.The same thing that happened in the OT also happen in the NT.For example you see side by side,verses from the OT and the NT,that show how wicked people die and go to hell.You can see them speaking(even though they are dead)and even look around them (even though they are dead),strange isn't it?
“Son of man, wail for the multitudes of Egypt, and consign both her and the daughters of the mighty nations to the depths of the earth with those who descend to the Pit:Ezekiel 32:18
Mighty chiefs will speak from the midst of Sheol about Egypt and her allies: ‘They have come down and lie with the uncircumcised,with those slain by the sword.’Ezekiel 32:21
Pharaoh will see them and be comforted over all his multitude—Pharaoh and all his army,slain by the sword,declares the Lord GOD.Ezekiel 32:31
Rich man in the NT
Notice how the rich man is in torment(even though he is dead),notice also,how he doesn't wish that anyone should come where he is(I think that somebody forgot to do his job,they should have told him that he is being refined for a short time[sad facepalm]).Notice how there is a great chasm between them,even if someone wanted to save him,they couldn't(We will let the one who is going to pick him up,after the refinement,figure it out).
The rich man also died and was buried.In Hades, where he was in torment, he looked up and saw Abraham far away, with Lazarus by his side.So he called to him, ‘Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.
“But Abraham replied, ‘Son, remember that in your lifetime you received your good things, while Lazarus received bad things, but now he is comforted here and you are in agony And besides all this, between us and you a great chasm has been set in place, so that those who want to go from here to you cannot, nor can anyone cross over from there to us.’“He answered, ‘Then I beg you, father, send Lazarus to my family,for I have five brothers. Let him warn them, so that they will not also come to this place of torment.’’Luke 16
Last verse “No, father Abraham, he said, ‘but if someone from the dead goes to them, they will repent.’ He said to him, ‘If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead.’”Luke 16
Conclusion:Jesus is the One who rose from the dead,if the rich had listen to Him(the Messiah),He would have turned from his selfish ways(by the power of the Holy Spirit)and and he would have gained eternal life.If Jesus come to anybody,they need to repent and pay attention to his teaching,for His words are Spirit and Life,and remember there is only one Teacher.
Nor are you to be called instructors, for you have one Instructor, the Messiah.Matthew 23:10