Why did people start calling this unnamed Pharaoh "Ramses" and when did they start doing it?
The answer is quite simple and short here: Bible kind of name-drops him in Exodus 1:11
Therefore they set taskmasters over them to afflict them with heavy burdens. They built for Pharaoh store cities, Pithom and Raamses.
While exact location or much about Pithom, Ramses II did build city called Pi-Ramsesses as his new capital. It's much harder to make connection between Biblical Pharaoh and any historical Pharaoh other than Ramses II.
I do not want to say "Bible says so, hence it's true", but it make perfect sense, especially if in connection with Joseph. His story is usually taken as having historical basis in Hyksos period, then Semitic invaders ruled Upper Egypt, after they have been driven out ("Now there arose a new king over Egypt, who did not know Joseph ") the Semitic semi-nomadic shepherds that settled Nile Delta become a distrusted minority that when was additionally taxed with forced labor in Ramses II's construction project, resulting in number of them leaving Egypt.
This theory lacks extra-Biblical evidence, but from what we have it's most reasonable one.