The name "Luna Park" traces back to Luna Park, Coney Island (1903) — the second of three iconic Coney parks (after Steeplechase, before Dreamland). Its creators, Frederic Thompson and Elmer "Skip" Dundy, chose "Luna" for its celestial, dreamlike allure.
"Luna Park" became a genericized brand for trolley parks worldwide (Melbourne, Sydney, Buenos Aires, Tel Aviv, etc.). The Luna Park Group (formerly Ted Gardiner Family) now operates several under trademark license.
"Park" simply denotes a dedicated recreation ground — standard for amusement parks, pleasure gardens, and public gardens since Victorian times.
So: Luna (Moon) + Park (Place of Play) = A place to visit the moon without leaving Earth.