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)
Copyright (c) 2012 Some Name
Copyright (c) 2016 Your Name
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
See also the answers to this question which is pretty similar.
Question @fbaierl, how would this apply if I wanted to add a different license to the repo? How would this work? Create a new file and maintain the original licence? Would this mean two licences in the same repo?