Apparently there are a few theories as to why the township was originally spelled with the doubled letters. Here's an explanation from OnlyMelbourne:
The discovery of gold in the upper Goulburn River and Woods Point areas in the late 1850s caused the formation of a miners' access track. The place where the Woods Point Road crossed the Olinda Creek was chosen for a town survey. The origin of the name is uncertain. One version is that the Government surveyor, John Hardy, suggested that the town be named Lillydale after hearing his chainman singing a popular song "Lilly Dale". The name was also inspired by the surveyor observing lilies growing in pools of the creek. The other version is that a district surveyor's wife was name Lilly. Council clerks and the local schoolmaster shortened the spelling to Lilydale.Given that the town was named in the 1850s, the song theory seems possible, as the tune was written in 1852 and was sung in America in the 1850s. Many Americans worked on the Australian goldfields.