This is my 6th version on the cup I have made for my hydroponic systems. My previous designs were all okay but all failed the same way, the pepper plant roots destroyed them. I beefed up the thickness and made sure the top was strong enough so I didn't have to worry about falling through my 5 gallon buckets but still small enough to be used with mason jars.
My method is cheap and simple. I germinate the seeds in a moist plastic bag, fill the hydroponic cup with clean (natural) aquarium gravel. and start them in mason jars. You just need to ensure the jar itself if complete blocked out from the sunlight (either using aluminum foil or painting the jars). Just make sure to use hydroponic plant food and de-chlorinate your water (I am using tap water right now but if you boil it or leave it out in an open container for a day most of the chlorine will evaporate)