Typically, vodka only adds a subtle flavor in cocktails, letting the other ingredients shine. With that as inspiration, the below products can be used in similar ways, where they still offer something, but are not the prominent flavor in the drink.
In the case of the top pick for non-alcoholic vodka, you even get some vitamins and caffeine added to your mocktail.
See the reviews below for mocktails made in the spirit of vodka cocktails, but with something new.