So many excellent choices on this list. Some of my favorites are listed below. I tried to only list those I haven't already seen on here, but I apologize if there are duplicates:
First a couple of classics:
Eleanor Rigby - the Beatles
Hallelujah - by about a gazillion people, but I love Jeff Buckley and kd lang's versions.
Mary Chapin Carpenter's Ten Thousand Miles:
https://www.youtube.com/watch?v=p8M8lssnf44
Jonatha Brooke wrote Time when her mom was in hospice. She compiled some songs into a play, but her song itself starts at about 1:15. I first heard it after my mom's death, and it can still make me cry.
https://www.youtube.com/watch?v=atvqubwXnrY