If I'm truly after "unique", by definition that means eschewing the normal "go-to" techniques and thinking outside the box. However, "unique" is rarely the goal. A better adjective might be "characterful", IOW not strictly unique but also not humdrum.
There is no effect that's off-limits. That includes effects normally associated with guitars and synthesizers, such as a Leslie emulator or a fuzzbox. Experimentation is encouraged.
But you asked for a specific technique, so OK, here's one: parallel distortion. Run the vocal through a distortion plugin, preferably one with a wet/dry mix control. A useful one that's also free is
Boogex from Voxengo. If you don't have a distortion plugin with a wet/dry knob, insert a send and place the effects on a bus. This has the added advantage of being able to effect the distortion component further, such as placing a chorus plugin after it.