Finally someone got it!
@Really Not All That Bright: Runny bunny is correct; and as far as I understand it, a hink-pink has one syllabe in each word, while a hinky-pinky has two.
How about a hinky pinky dinky:
An inexperienced pimp.
Nope, three syllables.
Rookie nookie cookie (or perhaps bookie)?
Confession profession?
A hinkey pinkey: A prime cut served with milk.
Yup–bookie! I realized after I posted it that “bookie” might not quite be the best descriptor, but I figured I see if anybody got it before offering a better clue. Good job! 
Yep!
Filet au lait
Another hinkey: bath time
shower hour (alternately “scour hour”)
Hinkey-pinkey: cloudy liquid suspension
Blurry slurry.
Hinkey-pinkey: Antelope religion
I’d say “shower hour” but it’s short a syllable.
Maybe the way you say “hour”.
Kudu Voodoo!
Hink-Pink: Deer execution device
Alright, I can live with that.
Here’s a hinkey pinkey: An alcoholic beverage from the streets.
Buckshot! 
(I know, I’m just kidding)
Seer beer?
Doe Bow!
Hink pink: container for smoked salmon
Lox box!
Ergh, wait, forgot to give one.
Hink Pink: Feline fight.