Can I split the mortgage interest payment on a tax return so that my co-owner and I file separately but still claim the house?

Yes, assuming you are itemizing.  

1. In the "Deductions and Credits" section select the "Mortgage interest..." topic in the "Your Home" dropdown.
2. Add a 1098
3. Enter the name of the lender
4. You will arrive at a page titled "Do any uncommon situations apply?"
5. One of the options there is "Someone co-owns this loan..."
6. You will be asked if "your social security number is on the 1098"
7. If yes,you will then enter your portion only of the form 1098 information
8. If no, you will first enter the information on the person who co-owns with you  
9. Then enter your portion of the 1098 info

 assumes the co-owner actually made part of the mortgage payments or they came out of a joint account.