Wow, you've posed a tough question.

As I'm sure you know, most make-ups and theatrical
colorings for hair are designed to be gentle,
temporary ones and therefore are water-soluble
(dissolved by water).

The only solutions that occur to me are drastic ones
to which I doubt your actor would be agreeable. And
if he does agree, he's opting for major abuse to his
hair and scalp.

1) The simplest choice is to have your actor's hair
professionally dyed. (Simple for me because I'm not
the one who'll be looking like Frank Drebin for three
months) However, most dyes are designed to cover
or remove gray hair -- I'm not even sure there IS a
permanent dye to make one's hair MORE gray. Any
beauticians out there who know better?

2) Another choice would be to use a product that is
not water-soluble and -- unfortunately -- the only
thing readily available would be white or gray spray
paint from your hardware department.   (Latex,
please, NOT oil-based!)   Once dried, the actor's hair
will be stiff, but you should be able to mat the actor's
hair with water. Unfortunately, the latex will make the
hair non-absorbent and much water will run down
the actor's face and shoulders.

Also it will take several, vigorous shampooings
afterwards to remove the gray so your actor can
resume his normal, daily life. Even then, removing
all the paint will be difficult if not impossible and your
poor actor will be chipping dried paint off his locks
for days to come. (If you've ever painted a ceiling ,
you'll know what I mean).

3) Most hair-gels and holding sprays are
alcohol-based and will dry the hair or make it stiff
and brittle. If you can find something that is
OIL-based (usually a more expensive, "natural" or
"organic" styling product) then the hair will remain
flexible and malleable.

The actor can apply a standard theatrical coloring (I
still think, penny-for-penny, white shoe polish works
best) and then CAREFULLY apply the oil based
styling product. (Oil and water don't mix -- if you apply
the oil first, the water-soluble coloring won't adhere)

Style the hair neatly for the beginning of the scene.
Then, when the actor leaves to "get wet" simply
muss or re-comb the hair to look matted. You may
need to re-apply the coloring, the hair-oil, or both
offstage (if you have the time).

Once again, clean up for your actor will be
inconvenient at best and at worst a nightmare-ish
"what have I gotten myself into?" rant.

4) And finally, use a common theatrical hair coloring
but use FAKE WATER. Unfortunately, the only
product readily available is glycerin which would
blow your budget or cooking oil (Yes, I said cooking
oil!) Crisco, sunflower, peanut, or what-have-you.
From a distance the audience can't tell if the actor's
hair is wet or extremely oily -- unless they can SMELL
him. In which case, not only will it ruin your
costumes, slick up your floor and make your actor
grumpy and messy, but it might just increase your
popcorn or muffin sales at concessions.

Best of luck.

Let me tell you....

Clown-White greasepaint, as recommended (I think) in one of the how-to articles here, will NOT readily wash out with just water...it requires soap, but I tried this just to see. It didn't run in water. It just sort of sat there, looking all sad and grey-whitish.