According to St. Bonaventure, the personal property that distinguishes the Father from the Son and the Spirit is his “innascibility” (innascibilitas), the fact, that is, that unlike the Son, he is entirely unbegotten (Breviloquium 1.3). As Russell Friedman has pointed out in his Medieval Trinitarian Thought from Aquinas to Ockham, this position becomes somewhat characteristic of the Franciscans generally.
Anselm seems to see things a little differently. In his Monologion, chapter 39, he writes that “it is the distinguishing characteristic of the second [person of the godhead] that is born from the first, and it is the distinguishing characteristic of the first [person] that the second is born from him” (Williams trans.). For Anselm, in other words, what makes the Father to be the Father has less to do with what he and the son do not share in common (the Son being begotten, the Father unbegotten), than it does with–in perhaps proto-Dominican fashion–with the relation that the Father and Son do share in common. The Son is the Son because he is begotten by the Father, and the Father is the Father because he begets the Son.