Green’s Dictionary of Slang

blue vein n.

also blue veiner
[the vein that runs up the penis]

1. the vagina.

[UK]‘The Blue Vein’ in Fanny Hill’s Bang-Up Reciter in Spedding & Watt (eds) Bawdy Songbooks (2011) III 318: When she pull’d up her clothes, Ned exclaimed, ‘I declare, / Your blue vein I can’t see, ’tis so covered with hair!’.

2. (Aus. prison/N.Z.) the penis.

[UK] ‘The Blue Vein’ in Fanny Hill’s Bang-up Reciter 26: Ned pull’d up his clothes, sir, when to their surprise, / They beheld his blue vein of a wonderful size.
[Aus]Tupper & Wortley Aus. Prison Sl. Gloss. [Internet] Blue vein. Penis.
[NZ]McGill Reed Dict. of N.Z. Sl. 27: blue veiner/blue vein flute. The penis ANZ.

3. an erection.

[US]J. Wambaugh Choirboys (1976) 36: Roscoe was unable to raise [...] a ‘diamond cutter’ or even a ‘blue veiner’ due to shooting pains in his groin.
[UK]G.F. Newman A Prisoner’s Tale 21: He [...] glanced down at his crotch. ‘I’m getting a blue vein here, Ginge,’ he said invitingly to the trusty.