This is a depressing question, but I'm not sure what I should/can do in this scenario.

My dad plays with my group every other week. But he got sick, and it doesn't look like he's going to be with us much longer. Obviously this is so incredibly far down my list of priorities I'm surprised I can even see it, but it's a distraction I can use that still makes me feel like I'm doing something useful.

Anyway. What should I do with his character? Should I continue him as an NPC, find someone else to play him (though I'm not sure I'll be able to handle that), write him out with a happily ever after story? Any ideas?

Honestly I may have to stop this campaign anyway, I'm not sure I can bring myself to keep going after this. This was a 100% homebrew and I created almost half the world with his character in mind - he'd never played before so I wanted to be sure I made it fun for him.  

So, yeah, any advice is welcome. So are distracting funny anecdotes, memes, cute baby animals, I dunno, I'm kind of just flailing around here. 

 There's no really good answer to this (I mean, it sucks, period), but fundamentally, when someone dies what you have left is memories, and if they're good memories, you should try to preserve them. In the short term, I doubt you'll want to continue the game (I would likely write a happily ever after for the entire campaign, not just the character, and have the entire thing as a memorial to player), but try to make sure it doesn't get forgotten, and in later times, after the pain has faded, you might want to bring it out again. Or you might not, but if you lose the memories, you'll probably regret it.

I agree with Pantagruel (as usual).

I wouldn't try to run that campaign, especially not right away. I don't think you'll be able to do it -- I know I wouldn't. 

True story... My father loved sports, especially baseball and football. When I was small I hated them, because they took my father's attention away from me. But then in 2nd grade he managed to get me interested in football (and a little baseball but I always liked football better). From then until I was 13, my father and I would sit together in the living room and watch the NY Giants, whom he loved and I despised (I was a Dallas fan, because I was a kid and hey, cowboys). The Giants were terrible in those days and I used to love ribbing him all game long, especially when they would break in to announce Dallas had won yet again (in those days, with Landry as coach, Dallas got into the post-season literally every year and won a couple of Super Bowls).

Then when I was 13, right as the football season was starting, my father passed away. He'd been sick for a long time (1.5 years) but it was still a shock. And after that, I found I just could not watch football anymore. I kind of followed the scores from week to week in the paper, but I could not sit there watching the game. And after that year I just stopped watching football entirely. I didn't watch football again regularly until Dallas had gone down hill and then came back and was having a good year, and this was like 10 years later, and I started watching it again. And it didn't bother me (and doesn't to this day, although I really don't follow sports at all anymore due to just not being interested).

My point is, for like 10 years after he died, I could not bring myself to do the thing I so associated with my father. Now, it would bring back great memories. But back then, it was too painful.

So... I guess what I'm saying is, you may find that you can't play D&D for a while. Don't beat yourself up about it. Find something else to do. One day, some years in the future, you will find that playing D&D brings you good memories of your father. But it won't happen right away.

Hopefully, though, this will all be a moot point and he'll make an unexpected recovery. My fingers are crossed for you. As I say... I know what it's like to be where you are.

As long as your friend is still alive, talk to him.

Find out what is important to him after he is no longer with us. If that includes anything he has planned for his character, try to enable that.

There is several stories about how games remember character of people who have died, and how it helps the community they belonged to to cope with the loss.

WotC for example has the Oblex, which was created for a Make-a-Wish recipient. Zipperon Disney has a fully funded Kickstarter to create a book in memory of Laurence, who died of cancer and wanted his character Galder to leave a mark in this world and in D&D.

I am sad to hear about your friend, and hope you can make his passing a little less painful. I also wish you the strength to cope with the grief, which is not going to be easy. Keep talking to your loved ones, it does help.

Everyone grieves differently, but it giving him the best eulogy and funeral in game might help you begin the healing process. Using the game as an escape from the situation seems like a very good idea as long as everyone else is able to understand that the game is not about the game anymore; its about your healing process. Best of luck and I wish your father a miraculous recovery.

My condolences.

Will he be able to play in another session or two? You could give his character a big send off. Beat the BBEG (and in this case fudge every single die roll if you have to) and let him ride off into the sunset. Or have his character become the new emperor, or just go to a quiet fishing village. Let him finish his character’s story on his own terms so he can have some control over something.
Or even just ask him about it in conversation, so you can have something distracting to talk about if you need that. 

Sorry if that’s a terrible idea, you know the situation much better than anybody else and I hope it doesn’t come across as callous, I’m just throwing out an option.

My dad introduced me to D&D almost 40 years ago; he died almost 30 years ago. We never played together (after a very short roll 3d6 in order type intro session), but I still remember him handing me that red box. It makes me happy. You’ll get there too one day. 

This reminds me of the story of Terminatur, the druid played by a player's grandmother who passed away while playing: https://www.dndbeyond.com/posts/585-a-grandmothers-d-d-character-that-will-never-be

According to his original Twitter thread, within the actual game the players continued without her character, but he planned to immortalize her character in future campaigns as an extra-planar NPC the players might not ever meet, but they'll at least hear stories.