Andreth did not prophesy. She quoted
Adanel, her kinswoman by marriage to Belemir of the
House of Bëor, whom she was visiting when
Finrod arrived.
Aredhel was the sister of
Turgon King of
Gondolin, the wife of Eöl the Dark
Elf, the smith who forged
Anglachel the Black
Sword of
Turin, and the mother of Maeglin, who betrayed Gondolin to
Morgoth.
Belemir was the grandfather of Emeldir,
Beren’s mother; Andreth’s brother was the grandfather of Barahir. So there are strong links to Beren One-handed in the Debate of Finrod and Andreth. (Barahir and Emeldir were already born at this point, and it is
possible, though definitely
not stated, that they overheard this discussion between Finrod and Barahir’s aunt, Andreth, as children.) Adanel was the sister of Hador, Chieftain of the Third House of the
Edain and grandfather of
Húrin and Huor, great-grandfather of
Túrin and
Tuor.
The “prophecy” in her debate with Finrod,
Ecthelion, is Andreth’s recitation to Finrod of the “Old Hope” of
Men that she had learned from Adanel: that
Eru himself would one day enter into
Arda and set aright the evils and marring of
Melkor Morgoth, a concept that shocked Finrod and one of which Andreth herself was uncertain.
I believe Andreth died before Túrin was born, though I could be mistaken about this. In any case, she did
not prophesy concerning Túrin: in fact, she did not (to my knowledge) prophesy at all, though both she and Adanel were considered “Wise-women” among the Edain.
The prophecy to which you refer is the Second Prophecy of
Mandos. That is the idea that the
Valar will grow inattentive so that Morgoth re-enters Arda, the
Númenóreans of
Ar-Pharazôn who were entombed under the
mountains of
Valinor in the
Akallabêth will be released to fight for Morgoth against the Valar, and Túrin Turambar will return with Anglachel and kill Morgoth in the
Dagor Dagorath, the “Battle of
Battles” at the end of the world, after which the
Ainur would sing together once more. It’s scattered across
Tolkien’s writings, and his concept of it changed over time.
Men, by the way, did not remain in Mandos. They left the Circles of the World and departed Arda. If you recall,
Lúthien kissed Beren after he was mortally wounded by
Carcharoth the Wolf of
Angband and bade him await her in Mandos: his soul refused to depart Arda until she arrived and pled their case to Mandos and
Manwë. That the
fëar, the souls of Men, leave Arda is
canon because in the “Tale of
Aragorn and
Arwen” in Appendix A of
Return of the King, just before Aragorn dies, he tells the grieving Arwen, “We are not bound for ever to the circles of the world, and beyond them is more than memory.”