You can have a look on my site
http://dicesites.com to see some of the popular dice sites with invest option. On there you can see how much the bankroll is and how much wagered the sites get. The more wagered a site has, the more profit it is expected to generate (in total numbers.) However, the bigger the bankroll is, the less share you get. Besides that, the house edge and commission are also relevant, which is currently not shown on my site.
However, I do like to emphasize the most important factor in choosing a dice site to invest in:
trust. There has been popular dice sites that run with all of the invested amount (Dice.ninja with more than 2.000 BTC) but also newer dice sites (Cryptocoins-dice.com with 150 BTC.) I personally do think that history of a site does establish some trust, but in the end it's up to you to make your judgments. If you send your BTC to some site (may it be an exchange or a gambling site), you do not control it anymore and they may scam you at any moment.