Heli of Luke's genealogy is the father of Mary

I hold that common view, but I don't argue it in the more common way that view is presented.  It's not about Heli being Joseph's father in law.  At least it's not only that.

In the Greek of Luke 3:23, first of all the order of words are different.  Secondly the entire genealogy in Greek only says "son of" once, and that's before the "as was supposed" which is put in parentheses.

In the Greek text of Luke's genealogy, every single name mentioned has the Greek definite article with one exception: the name of Joseph. Someone reading the original would understand by the missing definite article from Joseph's name that he was not really part of the genealogy.  I actually believe the parentheses in English should be expanded to include his name.

So I would render it.

And Jesus himself began to be about thirty years of age, being the son (as was supposed of Joseph) of Heli,

Thomas Aquinus also said that Luke says Jesus was the Son of Eli (Heli is a Hellenization of Eli) though he didn't explain why.  Thomas Aquinas, Summa Theologica, p. IIIa, q.31, a.3, Reply to Objection 2.

In both Greek and Hebrew the word for Son can mean Grandson or any descendant.

Tribal identity in Israel was usually determined strictly pater-lineally.  But in the case of bastards where the father was unknown, or a proselyte marrying an Israelite woman and fathering children by her, the pater-lineal tribal identity of the mother would determine the tribal identity of that offspring.

Also while inheritance can usually go only through sons, a man with no sons but daughters can pass inheritance through his daughters if she marries within the tribe.  Due to the law of the Daughters of Zelophehad, discussed in Numbers 26:33, 27, 36, Joshua 17 and 1 Chronicles 7.  Joseph being of the House of David would qualify as the same tribe.

Another example of a bad argument to use on this issue is the Talmudic reference sometimes cited as saying Heli was the father of Mary (Chag. 77,4).  This reference only seems to say this when a Christian is quoting it, so independent verification is wanting, and it has been cited by adversaries of this view as saying something different.  The Talmud is complicated to study, so maybe there is something there, but we're better off just avoiding it for now.

We should not build doctrine on Apocrypha, but the Infancy Gospel of James is interesting.  At first glance it would seem to contradict this view by clearly naming the father of Mary as Joachim.... But....

Joachim is a variant form of the name Jehoiakim.  There is only one Jehoiakim in The Bible, he was a King of Judah of the House of David.  The Infancy Gospel doesn't specify Joachim's ancestry, but other traditions related to his wife Anna/Hannah say she was of the House of David.  Tie that to him having a Davidic name and it seems he's Davidic.  [Update: I was rereading it recently and noticed the infancy Gospel does call Mary a Virgin of the daughters of David.]

King Jehoiakim was also known by the name Eliakim.  Those names not only look similar in how they end, but they have effectively the same meaning, just using a different name of God to communicate the meaning.  "He whom God has kept".  So it's probable any Jehoiakim could have also been called Eliakim, especially if a later one was named after the King.

Eli could very easily serve as a shortening of Eliakim.  And as I already said Heli was a Hellenization of Eli.

Yet the Infancy Gospel wouldn't have done it this way if it was seeking out to say the Biblical Heli is Mary's father.  While plenty in the Infancy Gospel is clearly false, being the prototype of Catholic Marian doctrines.  It may have been influenced by some real history that was passed on.