Green’s Dictionary of Slang

gigglestick n.1

[rhy. sl. (although not the rhy. sl. format of full phr.) = prick n. (1); the word may be equally placed among the various terms that equate penis with ‘weapon’ or laughter]

(orig. US) the penis.

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[US]Maurer & Baker ‘“Aus.” Rhyming Argot’ in AS XIX:3.
[US]Trimble 5000 Adult Sex Words and Phrases.
[US]Maledicta II:1+2 (Summer/Winter) 118: He [...] also lists: [...] Giggle Stick Prick.
[UK]S. Berkoff Decadence in Decadence and Other Plays (1985) 16: Fancy a tickle with the old giggle stick.