Subway did not begin as Subway, and its first name was not the founder's.
The chain began in 1965 in Bridgeport, Connecticut. Fred DeLuca was 17. Wikipedia says he borrowed $1,000 from his friend Peter Buck to open the shop. The Daily Meal says Dr. Peter Buck, as the sole investor, named it after himself: Pete's Super Submarines. Wikipedia lists the first name slightly differently, as Pete's Drive-In: Super Submarine Sandwiches.
"Submarine" simply named the sandwich on sale. The name pointed at the food, and no train came into it at first.
The name then shrank in steps. Subway's own history page says the name Subway was first used in 1968. The Daily Meal and Wikipedia's page on DeLuca also give 1968. Wikipedia's Subway article is inconsistent: its body says 1968, while its lead and infobox say 1972.
The steps in between are also uncertain. That infobox lists Pete's Subs from 1968 to 1970, then Pete's Subway from 1970 to 1972. The Daily Meal and the DeLuca page mention Pete's Subway as the stage before Subway. The Daily Meal says the change was made "in an effort to improve their marketing."
Both the Daily Meal and the DeLuca page repeat a popular reason for changing the name. DeLuca's New York accent reportedly made "Pete's Submarines" sound like "Pizza Marines" on the radio. The story is plausible and widely repeated, but neither source gives a direct quote.
Why "subway"? Nothing we read says who thought of it. It may have been a natural next clipping of "Pete's Subs," but that is a guess, and no source says a train was in mind.
The founders also formed a parent company in 1966, Doctor's Associates Inc. Wikipedia says the name reflects DeLuca's goal of paying for medical school and Buck's physics doctorate. The holding company's name is a separate story from the restaurant's.