How do I prepare Form 1310 if the court appointed my brother and I as beneficiaries of my mother's probate estate? The form only has one contact person.

Only one person can fill out the form, as only one person can claim the return for a deceased person.  The person named as executor/administrator of the estate is the one that will claim the refund.  If both of you are named the executor/administrator of the estate, then either one can claim the refund.   Do not send in two forms, as this will cause a delay in the refund.  

Thank you so much for your help! I just finished the return and it states I can't e-file it now. I guess I will have to print and send it in with Form 1310.  Your help was appreciated!

That is correct, you must mail Form 1310, either by itself or with the income tax return.  You will get the refund faster if you mail it with the income tax return.  

Perfect.  Thanks so much for your help!