Remember that the Muslims follow a lunar calendar... so a "moon" may mean a month (derivation of the word "month" comes from "moon").
Also, they consider a solar eclipse as complete proof of the END of a month. The new month does not start until the crescent of the new moon is seen.
Logically, a lunar eclipse, which occurs exactly halfway through a lunar month could therefore be called "splitting the moon (read Month) in two!"
Excelent! Makes perfect sense.
Maybe the "splitting of the moon" actually refers to what was revealed when the moon was split. In the texts, I believe that "Hira Mountain" was viewed between the two halves of the moon. Could this be significant?