Except for his grandparents Faramir and Éowyn, the details of Barahir's descent are uncertain. In The Heirs of Elendil (the ultimate basis for much of Appendix A), there is a single mention of a son of Faramir - apparently his only child - named Elboron. It isn't completely clear whether Tolkien meant this character to stand, but it seems reasonable to speculate that this Elboron was Barahir's father. This would in turn presumably see Barahir inheriting the titles of Steward of Gondor and Prince of Ithilien, though it should be emphasised that none of this is explicitly stated by Tolkien. (The Heirs of Elendil appears in The Peoples of Middle-earth, volume 12 of The History of Middle-earth).
Emyn Arnen, the hills of Ithilien, were granted to Barahir's grandfather Faramir as his seat, so it is reasonable to speculate that Barahir himself was at least raised there.
We have very little information about the descendants of Faramir, no more than indirect evidence for a single son, Elboron. If Elboron was Faramir's direct heir, and Barahir his heir in turn, then Barahir would have succeeded to the Stewardship and the title of Prince of Ithilien. However, we cannot rule out the possibility that Barahir was Elboron's younger son or nephew rather than his direct heir. In such a case Barahir would have been a minor noble rather than Prince in his own right.