How much BTC do I need as a bankroll to start a dice site? (Excluding development)
there is no perfect answer the more you have compared to the max win the safer you will be , but as devans stated 1% max win should be fine
it doesn't necessarily means that you should offer 0.5 BTC max profit , some sites offer lower than that like 0.1 or even lower
and you can of course run an investor based site when you basically take no risk at all but you will profit way less
Then again, for sites that do this, they have a LARGE issue attracting the best sorts of investors -- which are the whales that bet massive amounts of money. I would think a max min of 1 percent is fine, though if your bankroll is below 100 bitcoin I couldn't see you being able to attract too many whales. I don't think anyone wants to be betting for under 1 bitcoin if you're betting large amounts.
Bankroll is only one part of the site though, you're probably going to need a large amount of capital to have the site developed, company setup fees in an offshore island (depending on where you live, though you're probably going to need this), hosting fees and so on and so forth. You're probably looking at spending around 20k in original setup fees and such.
Not something which is going to be easy to do, plus you're going to have to be different then the REST of the competition - which there is many -- I do hope that whatever you decide to do works out. Goodluck!