If you changed your name in Second Life, all you need to do to update it in Bloodlines is to wear your HUD in-world, and it will automatically update your name. If it doesn't work, make sure you are using the most recent version of the HUD. When you wear your HUD, you will be auto-delivered the most recent version if there is an update available, but this will only happen a limited number of times if you don't start using the new one. If you don't have your previous deliveries of the most recent version, you can get it from the update terminal in any Liquid Designs Store, at Bloodlines East, Bloodlines North, Bloodlines South, Bloodlines Midland, or Bloodlines West.
If you are using the most recent version and it still doesn't update, try going to a different region and reattaching your HUD. We can only detect your name as it is reported by the region, and some regions will report your old name if it is cached there and the cache isn't updated with your new name yet.
For the support system, we can't automatically update your name, you will have to create a ticket and let us know, and we will update your name in the ticket system.