Is the solid chemical element mercury useful in outer space?

In what capacity? As a survival element? If you’re talking about mercury as fuel in space travel, it has been experimented with and has proven to be an effective additive to solid fuel. On its own, mercury isn’t a fuel (it could be if another source of energy is applied on it…but then, it defeats the purpose, doesn’t it) but as an additive to sold fuels and propellants, it helps in easier ionization. But since it is so toxic and since there are better alternatives, mercury isn’t used at all.
It would be interesting to note that ancient humans were fascinated with mercury and its properties. They experimented with it as a fuel source and have documented it too. Example: Ancient Indian Vimanas (aeroplanes) were supposed to have run on mercury based fuel….unsubstantiated of course….but it is a fascinating possibility….

I could see it being used in a sensor in its liquid state to confirm that a ship’s artificial gravity is functioning. Of course we don’t have this yet so maybe that’s not what you’re looking for.

I suppose you could vaporize it, ionize the vapor, then shoot it out the back end for thrust.

No. You would have to thaw it to use it, but then it would be too poisonous.