Prodigy: 1494, “sign, portent, something extraordinary from which omens are drawn,” from L. prodigium “sign, omen, portent, prodigy,” from pro- “forth” + -igium, a suffix or word of unknown origin, perhaps from *agi-, root of aio “I say”. Meaning “child with exceptional abilities” first recorded 1658. Prodigious.
is unrelated to…
Prodigal: c.1450, back-formation from prodigiality (1340), from O.Fr. prodigalite (13c.), from L.L. prodigalitatem (nom. prodigalitas) “wastefulness,” from L. prodigus “wasteful,” from prodigere “drive away, waste,” from pro- “forth” + agere “to drive” . First ref. is to prodigial son, from Vulgate L. filius prodigus (Luke xv.11-32).
...which really has nothing to with sons that go away, and don’t be confused by any closeness to…
Progeny: c.1300, from O.Fr. progenie (13c.), from L. progenies “descendants, offspring,” from progignere “beget,” from pro- “forth” + gignere “to produce, beget.”