Athena has similarities to the Furies, and they both go back to pre-Classical, Mycenean religion. I'm not sure there's a direct connection between them.
Athena is also a virgin deity, born of a mother alone (depending on who you ask), associated with snakes and the underworld (her son through Hephestios is Erikhthonios, who was a snake from the waist down), administers justice and is connected with the darker female powers through the Gorgon on her breast. (Her mother, Metis, gives her another link to the pre-Zeus female powers.) One of Athena's titles was axiopoinos, the avenger. The Furies also were worshipped in Athens, with a temple on the Hill of Ares.
The names Atana Potnija and Erinus (Fury,singular) appear in deity lists from Knossos, suggesting that both their cults are old ones,