A birthright was a right of possession or privilege often given to the eldest child.
In 2151, Captain Jonathan Archer, in discussing the Humans found on Terra Nova, described them to T'Pol as having the opportunity to return to Earth as "their birthright. It might take a little while, but they'll adapt. It's a hell of a lot better than dying down in those tunnels." (ENT: "Terra Nova")
In 2364, after admitting to Worf that they had destroyed a Klingon ship, Korris and Konmel stated that the Klingons aboard "had been corrupted by the illusion of peace" and that "they traded our birthright so they could die in their sleep." Later on, Worf noted that Korris had "talked of glory and of conquest and legends we will write," to which Korris pointed out that it was "the birthright of every Klingon." (TNG: "Heart of Glory")
Edit