It changes the meaning slightly (for me, when I'm thinking about it): Are you fine? Yes, I just have a bit of a headache. (Here, I only have a small headache.) Are you fine? Yes, I have just a bit of a headache. (Here, "just" and "bit of" both modify "headache," so there's no only as above, and the headache is even smaller.)
In the real world, however, both are pretty similar.
To be a headache is a perfectly normal idiom in BE, but it's got nothing to do with having a headache. I am a headache is on the same lines as I'm a pain in the arse; it means you are a source of trouble or worry to other people. People don't often say it of themselves.
More often you use the expression to refer to particular worries, rather than people, e.g. This new contract is a headache.