Desktop site > My Mumsnet (second from right on top of page) > My account (in the middle) > Username > type in the new name > enter password (down a little) > save changes (down a little more) and that should be you.
You used to get a coloured bar coming up at the top of the page to say you'd done it (I don't know what the new layout has) and you can always check by having a look on a shortish thread and seeing what name is in the message box.
Oh - and if it works, don't come back on here and say 'Yay - I did it' in the new name!. 