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From what I've been able to find, there is no ancient source which tells such a story about Aidepsos (the commonly older transliteration of the town's name, which is Latinised as Aedepsus). It seems that this "legend" has been cobbled together from a random grab-bag created by eclectically appropriating a variety of sources talking about Herakles in relation to baths or hot springs in general as well as specifically in relation to other places apart from Aidepsos.
Assorted References
According to Athenaeus' Deipnosophistai 12.512, all hot springs which rise up out of the ground are said to be sacred to Herakles.
Strabo's Geographika 9.4.2 says that there "are the hot waters of Herakles" at Aidepsos.
One of the characters in Aristophanes' Nephelai ("Clouds") asks, rhetorically, in Lines 1050-1052, "Where did you ever see baths of Herakles that were cold?" The scholion on this, in what has been labelled as Fragment 7 of Peisander (c. 600s/500s BC), tells us that:
At Thermopylai [Thermopylae] the goddess grey-eyed Athena made hot bathing-spots for
him along the edge of the sea.
From context the "him" is generally interpreted to be Herakles. This is referring, however, not to Aidepsos but to Thermopylai, which incidentally is roughly in the same neighbourhood but separated from Aidepsos by the Malian Gulf and several good stretches of land, with Aidepsos lying on Euboia [Euboea; Evia] Island and Thermopylai being on the Greek mainland. Hot springs are common in the region.
At this point Strabo does have more to say, in Geographika 9.4.13 (a little bit further down in the same chapter as the above-quoted passge), telling us that Thermopylai, meaning "Hot Gates," is so named because "there are hot waters near it which are honoured as sacred to Herakles".
Diodorus Siculus
The closest we get to the Edipsos legend in an ancient source is Diodorus Siculus, who refers to an event set in Sicily, quite some distance away from the Malian Gulf area(!). In his Bibliotheka Historika ("Library of History") 4.23.1, he says that:
Upon his arrival in Sikelia [Sicily], Herakles desired to make the circuit of
the entire island and so set out from Pelorias in the direction of
Eryx. While passing along the coast of the island, the myths relate,
the Nymphs caused warm baths to gush forth so that he might refresh
himself after the toil sustained in journeying. There are two of
these, called respectively Himeraia [Himeraea] and Egestaia
[Egestaea], each of them having its name from the place where the
baths are.
To go a bit deeper into the story we have to flip over to the next book, where, in 5.3.4, Athena becomes the patron goddess of Himera in Sicily. Here, the Nymphs, in order
to find favour with Athena, caused the hot water springs to erupt and flow forth vigorously when Herakles visited the island, and the inhabitants
consecrated a city to her and a tract of land which to this day is
called Athena’s.
The Souda
The only other information source to fall back onto is the 10th century AD encyclopaedia called the Souda [Suda; sometimes Suidae], which, for our purposes, helpfully fills in a few details.
- In the article
Herakleios psora, "Herakleian itch," it says that
this is a particularly difficult sort of itch to treat, requiring "Herakleian"
baths, "For Athena recommended Herakles many baths to cure his ills."
- According to the
Thermopylai entry of the Souda, "Some call this
city Pylai [Pylae, "Gates"]. But Philaias [Philaeas] says that it is
called Thermopylai ["Hot Gates"], since Athena made hot baths for
Herakles there."
- Finally, the article
Herakleia loutra, "Herakleian baths," says that this term means τὰ θερμά (ta therma,
"hot ones"), "For Hephaistos [Hephaestus] gave them to Herakles as a
gift. From these comes the term ta therma [the 'hot {i.e. hot
baths}']."
This last one seems to be the only reference anywhere to a link between Hephaistos and Herakles-baths. Athena is not mentioned in connection with him here.
Summary
As we see above, Diodorus Siculus appears to think it that was "the Nymphs" who were the agents of the creation of hot springs for Herakles on Athena's behalf, but the event does not even take place in Greece at all.
Geographically speaking the nearest we can get to the Edipsos legend mentioned by those websites is across the sea (and a decent chunk of Euboian promontory too) in Thermopylai, where none of our sources tells us anything about Athena requiring Hephaistos' assistance with her Herakleian baths project.
