Green’s Dictionary of Slang

1. (US) any small mechanical object [? SE dinky].

1912

R.W. Brown ‘Word-List From Western Indiana’ in DN III:viii 571: binky, n. Any little mechanical contrivance.

2. (drugs, also bopper) a needle used for injecting narcotic drugs, most often a disposable needle that is prescribed to a diabetic [bing n.¹ ].

1992		Bentley & Corbett Prison Sl. 80: Binky A type of modified syringe for the administration of narcotics. [Ibid.] 81: One of the original outfits used by drug addicts was the eye dropper fit […] From this type fit evolved the ‘binky’ or ‘bopper’ type fit employing the eye-dropper principle.
c.2003		Dalton Vrij ‘Tying Off’ on Inter-zone.org 🌐 Dalton had broken into doing dope in the era when the ‘Binky’ was still the rig of choice, the eye dropper barrel with the standard rubber part replaced by a baby pacifier for extra liquid leverage.