Abraham is first called an Hebrew in Genesis 14:13. Eber was the son of Salah, Genesis 10:24, who was the son of Shem. Shem is deliberately called 'the father of all of the children of Eber' in Genesis 10:21.
Balaam prophesied, Numbers 24:24, of a coming affliction (from a navy from Chittim) upon what he refers to as 'Eber'. Some have attached a significance to this regarding 'crossing over' but I have yet to discover why they do so. The overwhelming evidence is that 'Hebrew' means 'descended from Eber'.
I have counted thirty times in Genesis, Exodus, Deuteronomy, I Samuel and Jeremiah where the term 'Hebrew' is used in scripture. It appears to me to always have an ethnic meaning.
The writer to the Hebrews addresses 'holy brethren, partakers of the heavenly calling' who profess Christ Jesus (3:1) who are in danger of 'coming short' (4:2) who are 'dull of hearing' (5:11) who have certain who 'have the rule over them' to whom they are to submit (13:17). This looks like an ekklesia to me, that is to say a global thing not a local thing.The emphasis of content is clearly to ex-Jews.
So 'To the Hebrews' seems a fine enough description to my own mind as it denotes an ethnic characteristic (which still prevailed) rather than a religious title (Jews) which would no longer be appropriate.
The epistle clearly is applicable to ex-Jews who have been converted but the principles applied also apply to Gentiles, because they are spiritual principles which are common to all mankind, not just to those who were privileged to be of Hebrew birth.
I cannot see any connection with the concept of 'crossing over' in either the Hebrew scriptures or the Greek scriptures, myself.