There are plenty of names available that are somewhat related to unicorn, allowing you to keep the site design and pretty much everything else. Here are a few:

- Alicorn (what a unicorn's horn is made of, apparently)

- Oryx (An antelope-like creature that has a unicorn-ish horn and actually exists)

- Eland (another antelope-like creature that has two horns, but apparently is worshipped by tribes people)

- Unico (a cartoon unicorn)

- Peggy (another unicorn name)

- Camphurch (type of unicorn. ugly.)

- Ki-Lin (a chinese unicorn made of two separate unicorns)

Hope these help. Alicorn and Oryx are my favorites.