C.J. Dennis spent the latter part of his life in Toolangi, a small hamlet some 15 minutes from Healesville in Victoria's Yarra Valley. Since the early 1970s the town has been without a hotel after the previous one was destroyed in a fire. That hotel has now been rebuilt and a report in the Upper Yarra Mail states that the main bar will be given the title of "I Dips Me Lid" in honour of Toolangi's most famous resident.
See the poem "I Dips Me Lid" for an example of the usage. Of course, the most famous use of the phrase occurs in Dennis's poem The Intro, from his verse novel The Songs of a Sentimental Bloke. The relevant verse is as follows:
O' course we worked the oricle; you bet!
But, 'struth, I ain't recovered frum it yet!
'Twas on a Saturdee, in Colluns Street,
An' -- quite by accident, o' course -- we meet.
Me pal 'e trots 'er up an' does the toff --
'E allus wus a bloke fer showin' off.
"This ere's Doreen," 'e sez.
"This 'ere's the Kid."
I dips me lid.