Is it okay to befriend your significant other's ex online?

I would not do this. How can you know this woman will have your interests at heart? Your boyfriend and this woman broke up and they have moved on with their lives. It is wrong to talk about your boyfriend with this person. He deserves his privacy and she deserves to move on with her life. You would be smarter to seek the counsel of a wise friend, a clergy member or counselor if you want to gain insight on how to deal with your boyfriend.