Why apollo 11 name?

The name didn't come from a contest. Abe Silverstein, the NASA official responsible for spaceflight development, named both Mercury and Apollo, and with Mercury the precedent had already been set of giving crewed projects the names of gods and heroes from mythology. Apollo was the Greek god of archery, prophecy, poetry and music and, above all, of the sun, which he drove across the sky each day in his golden chariot: an image that gave the program the scale it was looking for. NASA publicly announced "Project Apollo" at the conference held on 28 and 29 July 1960.

The number is more mundane: it marks the flight's position in the list of Apollo missions, a list that was reordered after the 1967 tragedy. Mission AS-204, whose crew died in a cabin fire during a pad test, was redesignated Apollo 1 in honour of the astronauts. The first Saturn V launch, planned for November 1967, was then called Apollo 4, and no mission was ever designated Apollo 2 or Apollo 3. That is why the count jumps ahead, and why the eleventh name arrived sooner than the number of flights actually made would suggest.

It's worth not reading the eleven as a count of Moon-landing attempts: the eleven names don't correspond to eleven trips to the Moon, and the first of them never even flew. Within each mission, moreover, the crew gave its own call signs to the spacecraft: in July 1969 the command module was named Columbia and the lunar module Eagle, and it was those names, not the program's number, that were heard in the descent communications.