I've checked the TLG for both βερεκυντ- and βερεκυνθ- and PHI for berecynt-, which did not turn up a story about Heracles' son. I've checked Gantz, looked through any reference for Berecynt(h)us, and nothing.

Same with looking through Smith, but I should note that they don't actually cite Smith—or anything—for that information. Unsourced information on Wikipedia really ought to be ignored.

While it's possible that someone had access to some scholion or note somewhere that makes this connection, it's more likely that they just got confused.

My guess is that they misremembered this passage from the Ps.-Apoll:

and in a battle between him and the king of the Bebryces Hercules sided with Lycus and slew many, amongst others King Mygdon, brother of Amycus. And he took much land from the Bebryces and gave it to Lycus, who called it all Heraclea.

I also don't see this Astydameia mentioned anywhere, and for once the Wiki page on characters with this name is well-sourced.

This whole thing is strange, though, since there's little reason to connect Heracles to Berecyntus, the first priest of Cybele according to Pseudo-Plutarch, so I'm holding out the possibility that something exists, but it certainly doesn't stand within mainstream Greek thought, and it never made an impact on the Roman authors for whom Berecyntus is a mountain. It also is in line with Heracles as a progenitor of barbarian races, so I can imagine that someone somewhere connected the Phrygians to the Greeks via Hercules through this route, though they differ by using this niche toponym instead of e.g. having a son name Phrygus (see e.g. the sons of Heracles with the Scythian serpent queen in Herodotus).

I've checked the TLG for both βερεκυντ- and βερεκυνθ- and PHI for berecynt-, which did not turn up a story about Heracles' son. I've checked Gantz, looked through any reference for Berecynt(h)us, and nothing.

Same with looking through Smith, but I should note that they don't actually cite Smith—or anything—for that information. Unsourced information on Wikipedia really ought to be ignored.

While it's possible that someone had access to some scholion or note somewhere that makes this connection, it's more likely that they just got confused.

My guess is that they misremembered this passage from the Ps.-Apoll:

and in a battle between him and the king of the Bebryces Hercules sided with Lycus and slew many, amongst others King Mygdon, brother of Amycus. And he took much land from the Bebryces and gave it to Lycus, who called it all Heraclea.

I also don't see this Astydameia mentioned anywhere, and for once the Wiki page on characters with this name is well-sourced.

This whole thing is strange, though, since there's little reason to connect Heracles to Berecyntus, the first priest of Cybele according to Pseudo-Plutarch, so I'm holding out the possibility that something exists, but it certainly doesn't stand within mainstream Greek thought, and it never made an impact on the Roman authors for whom Berecyntus is a mountain. It also is in line with Heracles as a progenitor of barbarian races, so I can imagine that someone somewhere connected the Phrygians to the Greeks via Hercules through this route, though it differs by using this niche toponym instead of e.g. having a son name Phrygus (see e.g. the sons of Heracles with the Scythian serpent queen in Herodotus).