I got really frustrated with the woman on the phone who told me I needed to prove this, but had no suggestions when I asked her what I should send. She just kept saying it was my responsibility to prove residency. I had proof of Irish health insurance, and she was like, well, you could have moved here, set up health insurance, and moved away. My name was on the phone/electric bills. Same thing.
Have you been using the NHS? You can send records that you have been seen by the doctor or dentist. I sent in an entire copy of my passport and figured if they wanted to sort out all the stamps in there, they could have at it! I think I used rejection letters from jobs to prove I was attempting to find work. I printed out some of my blog to show pictures and stories (all with date stamps) I had written while living in Ireland.