I think it refers to contribute or help. For example, I do my bit for my city helping to keep it cleaned. It' is not a lot of help, but at least I'm doing something for it. 

Hi Anna! "We'll do our bit" is like saying, "We'll do our part." 

For example: You and I are planning a party. You say that you will bring food. I say, "I need to do my bit too. I'll bring drinks."

Hope that helps!