No. Sir John Franklin did not have a son. He had one child between his two marriages and it was a girl named Eleanor Isabella Franklin. Her mom was Eleanor Anne Porden Franklin. Hope that answered your question!
When Sir John Franklin finally docked near the arctic, he met some aboriginals. They traded their goods with one another (fur, guns, etc.) The aboriginals helped John Franklin with maps and navigating. They helped him and his crew find their way around the new land they had just stopped in.
Franklin is famous for his search for the Northwest Passage through the Canadian arctic in 1845. No one survived after both ships were stuck in the ice. See link below for the Northwest passage song that mentions Franklin.