The Greek word translated "mediator" is mesites, and one of the meanings of that word is "a medium of communication, arbitrator...used of Moses, as one who brought the commands of God to the people of Israel and acted as a mediator with God on behalf of the people" (John Henry Thayer, A Greek-English Lexicon of the New Testament, 401).
The Lord Jesus is the Prophet who served as the medium of communication between the Father and the people:
"I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth; and he shall speak unto them all that I shall command him" (Deut.18:18).
It was in His role of Prophet when He began to reveal truths which are essential to the gospel:
"Very truly I tell you, whoever hears my word and believes him who sent me has eternal life and will not be judged but has crossed over from death to life" (Jn.5:24).
The Lord's words there are in regard to the internal inheritance spoken of here:
"Therefore, He is the mediator of a new testament, so that those who are called might receive the promise of the eternal inheritance" (Heb.9:15).
The Lord also revealed this truth concerning His death:
"Even as the Son of man came not to be ministered unto, but to minister, and to give his life a ransom for many" (Mt.20:28).
During his ministry Paul received communications from the Lord Jesus in regard to His Last Will and Testament, the gospel. Here we read the following about how he received the gospel from the Lord Jesus:
"But I certify you, brethren, that the gospel which was preached of me is not after man. For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ " (Gal.1:11-12).
We also read Paul's words here which speak of further revelations which He received from the Lord Jesus:
"It is not expedient for me doubtless to glory. I will come to visions and revelations of the Lord...And lest I should be exalted above measure through the abundance of the revelations, there was given to me a thorn in the flesh, the messenger of Satan to buffet me, lest I should be exalted above measure" (2 Cor.12:1,7).
Therefore, it is in the sense of being a medium of communication that the Lord Jesus is described as being a Mediator of the New Testament.
John Chrysostom sums up the Lord Jesus' role as a Mediator, saying "How did He become Mediator? He brought words from Him and brought them to us, conveying over what came from the Father to us, and adding His own death thereto" (Schaff, Philip, ed., Homilies on the Gospel of St. John and the Epistle to the Hebrews by St. Chrysostom; Homily XVI [Grand Rapids: Christian Classics Ethereal Library], 736).