a little bit of both should mean "not much of either", if the phrase a little bit is taken literally. In some cases, though, the intended meaning of a little bit of both seems to be "a combination of both", with no specification of how much.Please give us context for your question: How and where you've seen or heard it used, or how you propose to use it.
I think I'd be more likely to ask for "a little of each." I actually can't think of a situation where I'd say "a little of both."Host: Would you like lemon meringue or sticky toffee pudding?
Panj: Could I have a little bit of both?
I expect my host to serve me with a smaller-than-normal helping of both. I wouldn't be surprised if this was two half-sized portions. But there's a good chance that it will be rather more than that