Haven't tried it with sea sickness yet, but I do have a very weak stomach and I buy ginger chews at Vitamin Cottage. I swear by them! They seem to work really well for me, taste pretty good (if you like ginger--they are pretty strong), they're CANDY, and they are natural. You can probably get them other places too. The ones I use are called Reeds; others will probably work just as well, as long as they're packed with real ginger. I saw a Myth Busters one time on the benefits of ginger and it actually worked really well in their experiment for motion sickness, though it may not work for everyone. Plus no drugs = no sleepiness (not sure whether Bonine causes sleepiness or not).
Up here they sell Ginger Gravol which seems to be the same thing. I've used it for motion sickness, stomach flu and hangovers and it works wonderfully.