I think that I would try mascara.  It would come out with eye makeup remover don't you think?

There is always Burnt cork!
Which was used by the Music Hall Minstrel performers, like Al Jolson!
From what I remember just light a candle & burn the end of the cork by rolling it in your fingers over the flame & apply the burnt sooted end to the roots. The cork is not hot at all, rub the soot in with the cork & it will also colour up the scalp as well!
You can just wash to remove it, even on the face it required either a light application of make up remover like liquid paraffin or cold cream, but I found soap &  water better. It requires more than just sweat or water to remove, I have even been outside of the theatre in the rain & it does not streak with a wet face.
It might be difficult to just colour up the roots with the burnt cork though, without getting it on the rest of the hair?
I never needed to try it on the hair but it just a thought!

How about using dark (appropriate to your hair color) creme shadow applied with a small piece of stipple sponge and/or a 1/4" angled brush?  Similar in concept to using white creme makeup for grey hair.  It should wash out fairly easily too, but I would test that first.

-- Tom

I just want to say thanks to everyone for their time.

You all gave great ideas! I'm going to experiment with a few

and report back as to which worked the best.