Does that mean she has snake eyebrows? Nose hair? Ear hair? Pubic hair?...

Blind people are her only option, but there's the open question if picturing her face after feeling with the fingertips triggers the effect. If yes, it's memetic and not her gaze.

Seems to be a bit of a worn-out joke at this point

Well, since the only myths we are given are not from her perspective, but her enemies', we can't say for sure, but consider:

You are a faithful priestess of Athena on a remote island. One day, the literal God of the Sea and Tempestuous Rage decides to walk into Athena's Temple and brutally rape you. Athena, the goddess you have so faithfully served, but who refused to protect her own priestess in her own temple from her own uncle, takes some serious offense at the sacrilege of her priestess being raped in her temple, and the Goddess of Wisdom decides that the only appropriate action is to punish the priestess by cursing you to never be able to be in the company of anyone ever again, and to add insult to injury and betrayal, you've been turned into a monster. Now, you have just been raped, cursed and transformed, and so you run away to an island. Now, when random men start coming to your island to "slay the vile monster", what do YOU do? The last time a man came to your island, he raped you and literally ended your existence as a human being, and the clear and stated intention of these men is to kill you. So yeah, you probably don't feel too bad about some dick-waving "heroes" getting turned to stone.

And when that one guy inevitably comes along with the help of Athena and chops off your head? Turns out that not only were you pregnant the whole time from the rape, but your children are born from the gushing wound from your beheading and are a flying horse and a giant.

Oh yeah, your head? Athena was so proud of herself for all of this shit that she made your severed, tortured death mask her personal symbol and stuck it to her fucking armor to intimidate her enemies.