No, you are not allowed to change the copyright notice. Indeed, the license text states pretty clearly:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
But you are allowed to add a copyright notice.
If you want to keep the MIT license, which is what I would advise you to do, you only need to add a single line to the license file, before or after Copyright (c) 2012 Some Name with your own copyright notice. The final LICENSE file will look like this:
The MIT License (MIT)
