As others have suggested, there are three ways to handle this:

1. Normal fares that are on AA.com.
2. Bereavement fares.
3. Award tickets.

You have to check all three to know which will be the best for any particular situation. If you suspect that this situation might arise, it is always best to keep enough miles in your account to have option #3.

In my case, we already had a ticket to visit my mother in two weeks when the nursing home called one evening to say she was fading fast. AA changed the tickets to the next morning, no change fee or fare difference. They offered to change the return also, but I asked them to leave it and nest in an award ticket, as we would need to go back to wind up her affairs and dispose of her personal property. They did that, and the only charge was the expedite fee for the nested award ticket. I was very pleased with how AA handled this.