A quick internet search reveals a situation in need of a term -- a person wants to learn the language of a distant ancestor, a language out of use in the family for generations. There is a term, "ancestral language," but it's used to refer to a linguistic ancestors rather than human ones, as in Proto-Indo-European being an ancestor of Hindi. "Recovered language" is usually used to mean languages brought back from the brink of extinction. "Heritage language" would seem an accurate way to talk about any language from someone's heritage, but in most of the recent literature it has a narrower meaning, specifically referring to a language spoken in the home, perhaps by the learner as a child, or by the learner's parents, and often imperfectly learned or later forgotten. It's often used in the context of minority languages or languages that are less respected by a privileged, majority community.
I propose the term "genealogical language." Yes, it's sometimes used in the way "ancestral language" is, but it seems to be a much less common phrase. It also has the advantage of evoking searching for one's personal past as an intellectual hobby -- whether you disapprove of such a thing, or think it's self-affirming, or just harmless fun.
It's possible (even likely) that someone else has thought of this, or that there is some other way of talking about this situation, buried in the tenth or twelfth page of the search engine. If so -- especially if it's bona fide linguistic jargon -- please let me know. In the meantime, I need a way of referring to this topic.