The content of the "hope" has been described in different ways from the beginning of the letter.
It is described in negative terms, as not falling under condemnation in the final judgement; "How shall we escape if we neglect such a great salvation?" (ch2 v3, RSV).
It is described in positive terms as being brought to glory (ch2 v10), and having everything under our collective feet (ch2 v8).
It is the promise of entering into the Lord's "rest" (ch4 v1), when we "cease from our labours as God did from his" (ch4 v10).
It is the "eternal salvation" which we receive through the Son (ch5 v9); that is, once again, being saved from the danger of being found guilty on the day of judgement.
I suggest that the references to "hope" really go back to the argument of ch6 vv9-12. The writer has just given a warning about unfaithfulness, but feels "sure of better things" in the prospects of his first readers. He urges them to show "earnestness in realising the full assurance of hope" and be like those who "through faith and patience inherit the promises". Surely this refers to the promises of salvation and glory and rest already mentioned in the previous chapters.
In the logic of ch6, v14 is a different promise being quoted in order to establish the point that God keeps his promises. This encourages us to seize hold of the main hope (v18), a hope which "enters into the inner shrine behind the curtain" (v19). In other words, it gets into the presence of God where Jesus has preceded us.
In fact I'm inclined to identify the hope with the promise in Thessalonians; "And so we shall always be with the Lord" (2 Thessalonians ch4 v17).