A short visit to Merriam-Webster online dictionary shows us that a cock can also be:

a device (as a faucet or valve) for regulating the flow of a liquid

I can see why that could be the basis for the name.

Additional ideas:

  • The color of a rooster, or being flashy like one.
  • Makes one cocky.
  • The word cocktail appears only around 1806 (M-W) – could it be related to a gun’s cock?

But why dillydally, dawdle, or delay. Let’s go get us some drinks!