From Rockets, Missiles, and Men in Space by Willy Ley, 1968 edition, pg. 390:
Except for GT-III the Gemini spacecraft were not given names as the Mercury capsules had been. GT-III did have a name; it was called the Molly Brown. The name was chosen by the command pilot of the flight, Virgil Ivan Grissom. Grissom (generally known as “Gus”) had had to swim to safety when his Mercury capsule Liberty Bell 7 sank after the second suborbital flight. At the time GT-III was being readied, there was a successful musical show on Broadway called The Unsinkable Molly Brown, and Grissom had decided that if a name could help to keep his second capsule afloat that would be the one. NASA officials were a bit reluctant to accept “Molly Brown” but when Grissom suggested “Titanic” as his alternative choice they gave in.