The context of Gal 3:20 is Paul’s argument that justification is by faith, according to the covenant that God made with Abraham, and not by works, according to the Law that was given through Moses. The OP asks: “What promise was given to Abraham?” Paul is referring to God’s promise to Abraham and to his seed in the book of Genesis.
“By Myself I have sworn, declares the LORD, because you have done this
thing and have not withheld your son, your only son, 17 indeed I will
greatly bless you, and I will greatly multiply your seed as the stars
of the heavens and as the sand which is on the seashore; and your seed
shall possess the gate of their enemies. 18 “In your seed all the
nations of the earth shall be blessed” – Gen 22:15-16 NASB
Based on God’s covenant with Abraham, Paul asserts that justification is by faith alone and not by works of the Law, and that it is faith in Christ which identifies the descendants and heirs of God’s promise to Abraham (v 7). Within this context, Gal 3:20 is a theme that is threaded throughout Paul’s arguments, one that contrasts the plurality/division of parties in the Law, against the oneness/unity of God.
Paul uses a legal analogy to make his points more clear, but legal concepts are themselves difficult to understand. The paradox in Gal 3:20 makes it even more of a riddle. Thus the OP’s quest for a simple answer may be too tall of an order to fill. In addition, Gal 3:20 is written in such a way as to allow for multiple interpretations depending on the context and Paul’s argument. While the New Living Translation cited by the OP’s is easy to understand, by locking Gal 3:20 into one meaning, it limits the text’s reach and power.
Returning to the OP’s question, the role of a mediator is to reconcile or bring two or more parties that are at odds together in an agreement. In this case, the parties consist of God and man. Given this context, there are two principal ways of understanding Gal 3:20 that correspond to the two covenants outlined in the text. In one, the mediator is Moses and the agreement is the Law. In the other, the mediator is Christ and the agreement is God’s covenant with Abraham.
When Gal 3:20 is viewed in terms of the Law, it points to the Law’s inability to bring about reconciliation. Paul writes, “Now, that no one is justified by the Law before God is evident; for, “the righteous one will live by faith” (v11). A mediator implies there are two or more parties who agree to comply with the terms that are set in the agreement. But if justification were based on the Law given through Moses, Paul argues that man would be incapable of meeting such terms.
“But God is one.” The same word εἷς (Strong’s 1520) is used by Jesus in reference to God (Mat 19:17): “There is only One who is good.” Paul appears to echo this sentiment in his argument. God alone is good, whereas all men are confined to sin.
v22 But the Scripture has confined everyone under sin, so that the
promise by faith in Jesus Christ might be given to those who believe.
Furthermore, if justification were through the Law, it would mean God was changing or nullifying the covenant that had previously been made with Abraham (v 17), which would in turn imply that God is two-minded. But God is one in the sense that He is unchanging, singular of mind and purpose (cf Num 23:19).
v17 What I am saying is this: the Law, which came 430 years later,
does not invalidate a covenant previously ratified by God, so as to
nullify the promise.
Thus far we have looked at Gal 3:20 through the lens of the Law, but the meaning evolves when viewed in terms of God’s promise to Abraham. Whereas a mediator implies the compliance/participation of two or more parties, the fulfillment of a promise depends on one party only. In God's covenant with Abraham, that party is God, and the mediator of His promise to Abraham is Christ, the seed of Abraham.
v16 Now the promises were spoken to Abraham and to his seed. He does
not say, “And to seeds,” as one would in referring to many, but rather
as in referring to one, “And to your seed,” that is, Christ.
Paul points to the fact that “seed” is in the singular. The promise was made to Christ alone, but faith is the means by which all are united to him.
v26 For you are all sons and daughters of God through faith in Christ
Jesus. 27 For all of you who were baptized into Christ have clothed
yourselves with Christ.
That God is the only party responsible for fulfilling His promise opens up another possibility for interpreting Gal 3:20, which is as a reference to Christ’s unity with God the Father. In other words, Christ is the mediator who, as both God and man, stands in the position of both parties to bring all sides together. Of the different connotations in Gal 3:20, this is the one that best resolves the contradictions in the text: "Now a mediator is not for one party only; but God is only one." Compare with the careful wording of John 17:22-23:
The glory which You have given Me I also have given to them, so that
they may be one, just as We are one; I in them and You in Me, that
they may be perfected in unity – Jn 17:22-23
Whereas Christ reconciles, the Law causes further division. The disagreements between Jew and Gentile Christians over the Law is after all the reason that Paul is making these arguments. Such division is in itself contrary to the Spirit. In contrast, Christ is the mediator who, in reconciling mankind to God, brings all parties together as one.
28 There is neither Jew nor Greek, there is neither slave nor free,
there is neither male nor female; for you are all one in Christ Jesus.
As a final thought, Paul's intent was not to invalidate the Law. Between the giving of the promise and its fulfillment lies a gap during which the Law plays a crucial role as guide and guardian until such time as faith would come (vv 24-25). If I had to restate Gal 3:20 in a way that a child might understand, I would say: God does not love us and bless us with His spirit because we are always good and obey the rules, but because He is one who keeps His promises to those who believe.