Etymology and original form obscure; the form -cocks, -cox appears earlier than -cock.
One suggestion is that the first element is prime, but though primecock is used by Florio, this looks rather like an etymological manipulation; other early writers held it for an alteration of Latin præcox ‘early, precocious’. Apparently the word was originally of slang or low use, perhaps somewhat obscene or equivocal.
His Supplication to the Diuell
Written by Thomas Nash, Gent
A New Letter of Notable Contents,
Printed by John Wolfe, 1593