Back then, names had great significance and meaning, so to know a person's name was to understand something about them. Giving a name to a baby was Serious Business and many Bible names are prophetic.
Jacob's name sounds like the Hebrew words for 'heel' and 'deceiver'. This relates to how he followed his twin brother out of the womb, grasping his brother's heel. And throughout his life, Jacob was a deceiver - till many years of struggling with God and man mellowed him. The episode in chapter 32 was Jacob wrestling all night with a mysterious character. At dawn the man touched Jacob's hip and wrenched it out of its socket (causing Jacob to limp thereafter.) At that point Jacob said, "I will not let you go unless you bless me."
Then the man asked, "What is your name?"
"Jacob".
"Your name will no longer be Jacob. From now on you will be called Israel, because you have fought with God and with men and have won."
"Please tell me your name" Jacob asked.
"Why do you want to know my name?" the man replied, then he blessed Jacob.
Jacob's request was an attempt to regain some control. But this theophany was controlled by God, who would not reveal His name, which cannot be had on demand.
With Moses, God chose to reveal His character by stating His covenant name. This would enable Moses to obtain the allegiance of the Hebrews in Egypt. They would want some evidence that Moses was truly God's representative. So Moses could claim to have the authority of God's name as his warrant.
The Divine Name can be translated as I am who I am, or I will be what I will be. It speaks of a God who is self-sufficient, self-existent, all encompassing, and without limitations, the one Being in the universe who is not dependent on something else for His existence.
Yahweh (the Tetragram) is a transliteration of the proper name YHWH and is a form of the Hebrew verb 'to be' so it could mean 'He causes to be' or, 'He who is', or 'I am'.
God is sovereign, and He is in complete control of revealing Himself to His covenant people. Over the centuries, God has revealed more of His character and, supremely, in the person of Jesus Christ, who is the exact representation of God, the visible image of the invisible God. "For God in all his fullness was pleased to live in Christ". - Colossians 1:15-19. That is why Jesus could say, "Before Abraham was, I Am." John 8:58