Tritogeneia
In today's Athena libation we read one of Her Homeric Hymns. The hymn includes the epithet Tritogeneia (Τριτογενῆ). After the ritual, one of the ritualgoers asked the meaning of this epithet. I had to admit that I didn't know. So I did some checking up. Apparently there are a lot of theories.
Here's what The Theoi Project has to say about it:
TRITO or TRITOGENEIA (Tritô or Tritogeneia and Tritogenês), a surname of Athena (Hom. Il. iv. 515, Od. iii. 378; Hes. Theog. 924), which is explained in different ways. Some derive it from lake Tritonis in Libya, near which she is said to have been born (Eurip. Ion. 872 ; Apollod. i. 3. § 6; comp. Herod. iv. 150, 179); others from the stream Triton near Alalcomenae in Boeotia, where she was worshipped, and where according to some statements she was also born (Paus. ix. 33. § 4; comp. Hom. Il. iv. 8); the grammarians, lastly, derive the name from tritô which, in the dialect of the Athamanians, is said to signify "head," so that it would be the goddess born out of the head of her father. (Schol. ad Apollon. Rhod. iv. 1310; comp. Hom. Hymn. 28. 4 ; Hes. Theog. 924.)
And here's the remarkably similar LSJ entry, via Perseus:
Τρῑτο-γένεια, ἡ, (γίγνομαι) Trito-born, a name of Athena, Hom. Il. 4.515, au=Hom. Il. 8.39, Hom. Od. 3.378, Hes. Th. 895,au=Hes. Th. 924, IG14.1389ii 1. (Variously expld. in antiquity, from the lake Τριτωνίς in Libya, from which an old legend represents the goddess to have been born, Eur. Ion 872 (anap.), cf. Hdt. 4.180; or from Triton, a torrent in Boeotia, Paus. 9.33.7, cf. Apollod. 1.3.6; or from a spring in Arcadia, Paus. 8.26.6; or from τριτώ, Aeol. word for κεφαλή (Sch.Aristoph. Cl. 985, Tz.ad Lyc.519; Athamanian acc. to Nic.(Fr.145) ap.Hsch.), i.e. head-born; or, born on the third day of the month, Ister 26 (the 23rd, τρίτῃ φθίνοντος, Sch.BT Hom. Il. 8.39); or, the third child after Apollo and Artemis, Suid. s.v. τριτογενής; or, as representing Nature, born thrice in the year, D.S.1.12; or because she was author of the three main bonds of social life, Democr.1b,au=Democr. 1b2=lr.)
II. the Pythagoreans gave the name Ἀθηνᾶ τ. to the equilateral triangle, Plu.2.381e; cf. τρεῖς, τριάς.
Here's what The Theoi Project has to say about it:
TRITO or TRITOGENEIA (Tritô or Tritogeneia and Tritogenês), a surname of Athena (Hom. Il. iv. 515, Od. iii. 378; Hes. Theog. 924), which is explained in different ways. Some derive it from lake Tritonis in Libya, near which she is said to have been born (Eurip. Ion. 872 ; Apollod. i. 3. § 6; comp. Herod. iv. 150, 179); others from the stream Triton near Alalcomenae in Boeotia, where she was worshipped, and where according to some statements she was also born (Paus. ix. 33. § 4; comp. Hom. Il. iv. 8); the grammarians, lastly, derive the name from tritô which, in the dialect of the Athamanians, is said to signify "head," so that it would be the goddess born out of the head of her father. (Schol. ad Apollon. Rhod. iv. 1310; comp. Hom. Hymn. 28. 4 ; Hes. Theog. 924.)
And here's the remarkably similar LSJ entry, via Perseus:
Τρῑτο-γένεια, ἡ, (γίγνομαι) Trito-born, a name of Athena, Hom. Il. 4.515, au=Hom. Il. 8.39, Hom. Od. 3.378, Hes. Th. 895,au=Hes. Th. 924, IG14.1389ii 1. (Variously expld. in antiquity, from the lake Τριτωνίς in Libya, from which an old legend represents the goddess to have been born, Eur. Ion 872 (anap.), cf. Hdt. 4.180; or from Triton, a torrent in Boeotia, Paus. 9.33.7, cf. Apollod. 1.3.6; or from a spring in Arcadia, Paus. 8.26.6; or from τριτώ, Aeol. word for κεφαλή (Sch.Aristoph. Cl. 985, Tz.ad Lyc.519; Athamanian acc. to Nic.(Fr.145) ap.Hsch.), i.e. head-born; or, born on the third day of the month, Ister 26 (the 23rd, τρίτῃ φθίνοντος, Sch.BT Hom. Il. 8.39); or, the third child after Apollo and Artemis, Suid. s.v. τριτογενής; or, as representing Nature, born thrice in the year, D.S.1.12; or because she was author of the three main bonds of social life, Democr.1b,au=Democr. 1b2=lr.)
II. the Pythagoreans gave the name Ἀθηνᾶ τ. to the equilateral triangle, Plu.2.381e; cf. τρεῖς, τριάς.