Nylon can be dyed with acid dyes, which are the same type of dyes that work on wool, or with disperse dyes, which are the only kind of dye that works on polyester. Acid dye is better for nylon because it is more washfast (wash resistant) than disperse dyes, on nylon.

All-purpose dye, such as Rit, contains acid dyes in addition to direct dyes for cotton, so it can work on nylon. Unfortunately, the kind of acid dyes used in Rit is not very good. It tends to wash out and fade faster than the best acid dyes. If you're in a hurry and don't want to wait to mail-order a higher quality dye, though, Rit is a good choice.

To get good results with Rit dye on nylon, you must follow the right recipe. It will not work in cool water! First dissolve the dye in a pot of water and add white vinegar, about 100 ml per four liters (or 7 tablespoons per gallon). Use a large non-aluminum cooking pot which you intend to never again use for food, since Rit dye is not safe in food-use containers. Add your shirt to the pot of dye and heat, stirring constantly, until the dyebath reaches a low simmer, no more than 185°F (85°C). Keep the dyebath at this temperature and continue to stir constantly for half an hour or an hour, or less time if the fabric takes the color very quickly or if you want a paler color. Allow the nylon to cool in the water, then wash it very thoroughly in cool water. To launder, always hand-wash it separately, in cool water, because Rit dye tends to bleed in the laundry.

The very best dye for nylon is Lanaset dye; WashFast Acid dyes are also good. These dyes can be purchased only by mail-order, unless you have an exceptionally good crafts store nearby. These dyes will last longer on the fabric than Rit dye , and they can be washed in warmer water. The dyeing method is the same for all acid dyes.

See "How to dye nylon:" http://www.pburch.net/dyeing/FAQ/nylon.shtml

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[This answer was first posted, by me, on Yahoo answers, on November 9, 2008.]