Who Was Joseph's Father?

I was asked today about the seeming disparity between the genealogy of Matthew and Luke, both of whom provide a different father’s name for Joseph the (supposed) father of Jesus: Jacob (Matthew 1.16) and Heli (Luke 3.23).
There are two basic proposals:
1) That both genealogies refer to Joseph, with Matthew’s account intended for Jesus place as heir to the throne of David and Luke’s account intended for the actual biological lineage of Joseph.
2) Matthew is recording Joseph’s genealogy and Luke is recording Mary’s. This is supported by numerous early Fathers: Origen, Irenaeus, Tertullian, Athanasius, and Justin Martyr.
It has been suggested (in support of the second proposal) that Mary’s genealogy is given under the name Joseph (by Luke) because (A) women were not official heads in the genealogical records of the ancient world, though they could be mentioned (such as in Matthew) it was always in connection to a husband/father, and (B) that perhaps Mary was an only child (speculation, I know) and would be the family inheritor whose husband is then adopted as the heir for her. Under the second explanation it is usually pointed out that this would make Jesus the heir of David (and Abraham) by both adoption (through Joseph) and by birth (through Mary).
What are your thoughts?