willy

From Middle English willy, willi, equivalent to will +‎ -y. Cognate with Dutch willig (“obedient, hearsome”), German willig (“willing”), Swedish villig (“willing, agreeable”).

Originally northern British usage, from the 1960s. Probably the simple use of a proper name as a pet name; compare dick, fanny and peter. Unlikely to be a contraction of Latin membrum virile, male member (that is, the penis), a Latin term used in English in the 19th century.

Perhaps a shortening of willy-nilly, an old way of saying “whether one wants to or not”, here referring to the person being used as an agent without their knowledge, i.e. regardless of their will.