Luckybit doesn't have an investment option since it started in 2013, but i heard there is under development an affiliate program, i know that is not the same as invest but it would be nice to let users win from their referrals.

If you have a big amount of bitcoin and would like to become an investor you can try to contact the LB support, If you only want to invest 0.001BTC then is a waste of time for all.