There is no set number that is correct. You will find the largest and most successful on line t-shirt companies have more shirts then anyone wants to count. They also have the resources to do that. You're deciding factor will be money. How much do your blanks cost, what are your printing costs, art costs. Certainly the idea of 6-12 is great if you can do it. If you can not, launch with whatever you can afford to do. Make some sales and grow from there.
You will find that no matter how many designs you offer, some people will love them, some will hate them and most people will be indifferent. Everyone has their own tastes and you can not argue with people who like something different then you. They are just as right as you are.
Shortees custom manufactures our shirts so we can not just order up a few dozen blanks and try a new design for a relatively low cost. We have to carefully manage our inventory and because of that we have limited design options at any given time. I get tons of slack over the designs and the people who don't like them are entitled to their opinion and they are correct whenever they want to express it. If you are using commonly available blanks it will be easier for you to print small batches and try different things. Experiment a little and don't be to frustrated by the results if they are not great right away.
Just launch with a few to get your feet wet. Don't worry about the number. Even 4-6 will work if they are good designs and potential customers can find your store and buy.