There doesn't seem to be a very deep mythological reason for this. The "Apollo" name was decided in 1960 by a committee at the Lewis Research Center (now known as the Glenn Research Center at Lewis Field), where the largest parts of whole program were planned, including the development of the liquid hydrogen propulsion system.
Specifically the name was conceived by the director of the center, Dr. Abe Silverstein, in early 1960, because he thought the image of "Apollo riding his chariot across the Sun was appropriate to the grand scale of the proposed program."

Shortly before his death in 2001, when he was asked for the reason why he chose Apollo as the name in an interview with the New York Times he replied with "'No specific reason for it, [...] It was just an attractive name.'"

Source:
Apollo 11 Press Release, 13. July 1969
http://www.nasa.gov/centers/glenn/about/history/apollo_press_release.html

"Abe Silverstein, 92, Engineer Who Named Apollo Program", Wolfgang Saxon, New York Times, Published: June 5, 2001
http://www.nytimes.com/2001/06/05/us/abe-silverstein-92-engineer-who-named-apollo-program.html?pagewanted=1

Apollo was part of NASA's manned spaceflight program. Mission names (Apollo, Gemini, Mercury) were about manned spaceflight in general, and not chosen because of the destination of a particular mission.

Abe Silverstein, Director of Space Flight Development, proposed the name "Apollo" because it was the name of a god in ancient Greek mythology with attractive connotations and the precedent for naming manned spaceflight projects for mythological gods and heroes had been set with Mercury. Apollo was god of archery, prophecy, poetry, and music, and most significantly he was god of the sun. In his horse-drawn golden chariot, Apollo pulled the sun in its course across the sky each day.

Emphasis mine, you can see why this name would fit putting astronauts in a rocket (chariot) and have them travel a very long distance, such as the moon.

For example, the name Gemini was agreed upon, amongst other reasons, because it means twins in Latin, and the Gemini spacecraft had a two man crew.

You can read about the missions and chosen names in more detail at history.nasa.gov SP 4402