"Now the God of peace, that brought again from the dead our Lord Jesus, that great shepherd of the sheep, through the blood of the everlasting covenant," (Hebrews 13:20)
Notice that the 'everlasting covenant' in the Bible is one made by the blood of Jesus in relation to his sheep (his own true believers), for salvation. In the Genesis story of the bow in the clouds, this is also said to be the token of the 'everlasting covenant': "And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth." (Genesis 9:16) 'Every living creature' is Bible lingo for those persons that God is saving unto eternal life. That is who the eternal covenant is intended for. Romans 11:27 describes the covenant as the taking away of sins: "For this is my covenant unto them, when I shall take away their sins."
The bow is a Bible word describing Jesus, basically. This can be seen in places like Eze 1:28: "As the appearance of the bow that is in the cloud in the day of rain, so was the appearance of the brightness round about. This was the appearance of the likeness of the glory of the LORD. " Rev 14:14 shows Jesus associated with a cloud: "And I looked, and behold a white cloud, and upon the cloud one sat like unto the Son of man, having on his head a golden crown, and in his hand a sharp sickle." Clouds in the Bible are often pictures of judgment and of God's glory. When we look at the bow in the clouds, I think it is most likely intended to be saying we look upon Jesus at the cross where God poured out judgment and glorified Christ for the sake of the salvation of his own. When we look there, it is a token of his everlasting covenant between himself and true believers, that he would not cover them with judgment (the flood is a picture of judgment).