It doesn't matter who is on the mortgage, except that if your father is the only borrower, the mortgage will likely need to be paid off or refinanced by whoever receives the house.
What matters is how the house is owned. If your father is the sole owner of the house, then his Will can leave the house to you. Your mother would have the right to try to make a claim for year's support against the Will, if she and your father are still legally married at his death, but if your father wants to protect against that possibility (I assume that your father is still alive, based on the way you asked the question), there are ways he can do so. He should get an experienced estate planning attorney to help him with the Will.
Now, if your mother and father own the house jointly, as "joint tenants" (which mean the deed either specifically says that they own it as joint tenants or at least contains language that makes clear that the surviving owner receives the ownership of all of the house automatically at the first owner's death, then no, your father cannot leave you the house under his Will IF your mother survives him. That's because the right of survivorship, in that case, would override the Will. But if they own the house as joint tenants and he survives her, then his Will could leave the house to you.
If they own it jointly, but NOT as joint tenants (meaning that the deed lists both of them as owners but does not say as joint tenants or otherwise make clear that there is a right of survivorship), then your father's Will can only control his interest (1/2, if there are only 2 owners, unless the deed says some other share). He could leave it to you. Again, your mother could try to claim it as year's support, if she is legally married to your father at his death.
I hope this helps. Ideally, as I already mentioned, your father will consult a good estate planning attorney to help make sure his wishes are actually carried out.
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