mum or Mum?

I've been writing something recently from the POV of a young girl who always refers to her mother as 'mum'. When I write dialogue in which the girl addresses her mother I find myself capitalising 'mum' as though it's a name, which, in a way, I guess it is. For example 'I know that, Mum,' I said.

Even if she's referring to her when talking to other people I find myself writing 'Yeah, that's what Mum said' (or somesuch).

I have no idea what the correct way of doing this is but it feels wrong not to capitalise. I've asked a few people and, so far, everyone says it's better to capitalise. What do you think?

Tweet me (@matthwrites) and cast your vote.

Matt