Practically every small detail in this little narrative of humble beginnings has a reference to something else, and generally something much greater. And the details are interconnected, so we will deal with them in a single long answer.
(1) The prophecy of Joseph’s name, He Will Add. Rachel’s midwife said, “Fear not; thou shalt have this son also.” (Gen 35:17) This is a prophetic reference to the very name of Joseph, which meant He Will Add, and God does indeed add a son, Benjamin. But later, the Lord “will add” more than Benjamin, and at Bethlehem, which is, of course, the birthplace of both David the king and Christ the king; and if Benjamin was the least of the twelve Israelite patriarchs, David was the greatest of the merely human kings, and Jesus was the greatest of all kings and indeed of all beings, insofar as he is substantially the same as God the Father. So he did add, indeed.
(2) The recurring, prophetic Bethlehem connection. We just saw the recurring significance of Bethlehem, as the birthplace of David and Jesus. Indeed, of all the places in the Promised Land where one final child of Jacob could be born, it was especially apropos and prophetic that Benjamin should be born here.
The historical significance of Bethlehem as the place where Rachel became a mother of the Israelites is supported by the book of Ruth, even before David was born, when the elders of Bethlehem told Boaz:
The Lord make the woman that is come into thine house like Rachel and like Leah, which two did build the house of Israel; and do thou worthily in Ephratah, and be famous in Beth-lehem... (Ruth 4:11)
(3) The prophetic meaning of the city names, the House of God and the House of Bread. The family has just traveled from the House of God (Beth-el) to the House of Bread (Beth-lehem). This is deeply significant given what we have noted in (1) and (2). Through common and special grace, God provides “our daily bread” (Matt 6:11); and the greatest bread he has given us is the “Bread of Life” (John 6:48). It so happens that the person who taught us to pray for “our daily bread” was this “Bread of Life”—born at the House of Bread.
(4) Out of death, life—again. Benjamin, sadly, was born as his mother gave birth to him, and thus life came out of death. Jesus, too, died so that we might live, and his own resurrection from death to life was the first illustration of the hope of man he gave us.
(5) Son of my Sorrow and the Man of Sorrows; Son of my Right Hand and he who sits at the right hand of God. In this connection it is especially striking that Rachel proposed to call the infant Ben-oni, Son of my Sorrow, considering that Jesus was himself called the Man of Sorrows (Isa 53:3). Yet more astonishing is the fact that Jacob, the infant’s father, decided to call his son, instead, Ben-jamin, Son of My Right Hand, while God the Father of Jesus determined that he shall “sit on the right hand of the power of God.” (Luke 22:69).
Indeed, the meaning of Benjamin’s name refers to nothing in his own life or the life of his descendants, as near as I can tell; it seems much better explained in this prophecy. Neither he nor the tribe of Benjamin really serves as the “right-hand man” of either Israel the man (Benjamin's father) or Israel the nation (indeed, Benjamin was disgraced and nearly wiped out in Judges 19-21). That is, insofar as the meaning of the names “Ben-oni” and “Benjamin” are prophetic, the prophecy does not seem to point to Benjamin the man or the tribe, but rather to the famous seed of Israel—the Messiah most importantly, and the nation of Israel as “the right-hand man” of God among the nations.
In short, the birth of Benjamin at Bethlehem, upon his mother's death, most significantly prefigures, and is a type of, the birth of "the seed of Israel," especially the Messiah who had his start in Bethlehem. But it has other interesting prophetic connections as well.