It may not specifically state, "so-and-so gave such-and-such name to God," but the Bible is pretty straightforward that others before Hagar named God.
As to why it bothers to mention this instance, it could be for several reasons, all of which would just be speculation. Genesis 16 mentions this took place by a specific (and presumably well known at the time of the writing of Genesis) well, and at the end of the chapter, we see that it is explaining how the well got its name - it is named after this exchange between God & Hagar. So, it could just be pointing out historical context.
It could also be a simple lesson about God caring about everyone, not just Israel. He may have been "the God of Israel," or "the God of Abraham," and Israel may have been God's "chosen people," but here God is pointing out that He is also the God of everyone, that He's not above showing love & mercy to those other than "the chosen people".
It could also be attempting to show that Hagar wasn't a bad person, she was more just a victim of circumstance. I think people tend to look at Hagar & Ismael as "the bad guys", and this is pointing out that it wasn't really the case. Hagar experienced a closeness with God that most Jews & Christians don't ever experience in their lives.
It could be one of those, it could be all, it could be none.