The Greek word for a mediator is used of Moses (Gal. 3:19). Mesitēs means one who stands between covenanting parties bringing them into fellowship or forming a channel of communication between them. Moses was the mediator of the old covenant law. Jesus came as the mediator of a new covenant that superseded the old and broken covenant.
Mediator of a new covenant in Greek:
Diathēkēs neas mesitēs (Heb. 12:24)
Diathēkēs kainēs mesitēs (Heb. 9:15)
Strong’s Concordance numbers: 1242, 3501b (neas), 2537 (kainēs), 3316
Bible references: Heb. 9:15; 12:24
For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant. (Hebrews 9:15 NIV)
In Hebrews 8:6, the NASB wording suggests that “Mediator of a better covenant” is a name of Jesus. However, the Greek does not support it and the NIV wording is superior.
Elsewhere, Jesus is simply called the Mediator.