Yes you can but it cost's a fair bit of money, I think it's in the region of £150 to change a name, I enquired once as my horses BSJA name is different to his BE/BD name and I wanted them all the same, but not enough to pay £150 !!!
When I looked it up when I bought my mare it was going to cost £175 to change her name
so I kept it as it is and am now slowly getting used to it and even using it at unaffil shows where I could technically call her anything lol!
ETA: http://www.bsja.co.uk/fees.html This page has it on under other fees - then change if name, is definately £175!!
i 'think' you will find its actually free now to change a showname with the BSJA after all the recent hoo ha of names being spectator unfriendly as too difficult to pronouce. Ring the BSJA and they will confirm the cost or lack of it!!