I would say that Limbo is a decent candidate.

Sorry to hear things have been rough for you. I hope they get better soon. Here are the games that made me sad but also made me want to continue to the end to feel the feelings:

That Dragon, Cancer (not on GOG) This was a very strong cathartic experience for me when I played. It's about a family dealing with their baby's cancer. There are a few annoying minigames, but for the most part, man, it hit me hard.

Brothers: A Tale of Two Sons (not on GOG) I see some others have recommended this, and I do too. This is about two boys who go in search for medicine for their father. As they get farther from home, things get more surreal, huge, and overwhelming. It does interesting things with its controls.

Papo e Yo (not on GOG) A boy has a giant monster friend who gets violent when he eats a certain food. This is a beautiful puzzle-platformer with authentic Brazilian street art based on the author's experience of living with an alcoholic father.

Human Resource Machine Gameplay consists of programming, but the theme is of working at a meaningless office job 9-5 with only the hope of retirement to drive you to continue.

Dropsy You play a mentally-deficient clown who has been accused of burning down the circus tent with his family inside. Point-and-click puzzles to try to get people to stop hating you.

Fran Bow You are a young crazy girl dealing with frightening imagery. Point and click to nevertheless go on despite your mental state.

Massive Chalice A turn-based strategy game about leading a centuries-long war against a horde of demons. Your fighters grow old and die and are replaced by the next generation. It plays similar to XCOM but with a more serious tone and also (you may not think this an improvement, but I did) none of the endless cut-scenes interrupting the battles.

Shelter You play a mother badger leading the kits on a journey. You must keep the offspring safe and fed, and you will fail.

Unrest A minimalist RPG. You play a series of characters at key decision points of their lives. The options they have are all pretty bad (for example, the first character is a young girl being forced into an arranged marriage).

Reus In this game, you have three or four elemental giants that can change a planet. You try to get combos of things near each other to spawn improvements. If you are me, the worlds you try to create will get better to a point then spiral out of control and end in death and extinction. But you may actually good at the game, so results vary.

Din's Curse This is a procedurally-generated action RPG where you have been a terrible person, so in your afterlife you are condemned to fight as a champion of the god, Din and try to save a series of towns that are full of poverty, thieves, and monster attacks.

The story of The Lost Files of Sherlock Holmes: The Case of the Serrated Scalpel is pretty sad too. Without spoilers: it's about noble needs that lead to disasters if you pursue them at all costs...

Max Payne 2 is, I think, a relevant choice. Whereas the first game talks a lot about the loss of loved ones, it was more about revenge than coping with it. The second is more focused on the aftermath and how to decide on what's next.

A surprise await you if you finish it again on Dead on Arrival difficulty.

I don't know what to say, may time appease your wounds.

Much ta for the LOTS of input suggestions and well-wishes! Ok, at 75% off currently I'll give them a try. This one's been on my radar for ages. I see it's available DRM-free too. I actually have To The Moon. That Dragon Cancer is Steam only? Couldn't find it anywhere else. Have it. Thanks for the suggestion, will try it out. And I won't run out of depressing music easily, ex-goth that I am ;-) Looks good. But it has no free saving? Then it's just not possible right now.

I've been eyeing this one for ages. But having been Dark Eye DM in PnP for ~20 years I can't imagine playing a Dark Eye game in English. Will certainly pick it up once Daedelic stops being a bitch and delivers the German version to GOG.

I'd love to play Bioshock Infinite, but Steam... And both Ico and SotC are games I'm really eager to play, but they're quite hard to come by (lacking a PS I'd need a disk version to use with an emulator).

I don't know if Fran Bow will be a bit too disturbing for me right now. Syberia is an excellent suggestion, and it's been sitting on my shelf for ages. Installing now.

Kholat looks great. I just can't afford it right now (we're moving from the city to the countryside and the moving service and the paint job for the old place will eat up any saving we have).
I've actually played The Vanishing of Ethan Carter. I really loved the atmosphere, but the save system is just incompatible with my life.

The download link on that site leads to a domain parking page. Got a mirror?

It's a great game too, so that's a bonus :)

Yeah, probably not much I can recommend that you don't already have then ;)

Same here. Happy cheery stuff either creeps me out or makes me irrationally aggressive atm.

Braid... have it on disk, it's already packed up in the removal boxes. Has to wait.
Whispered World...too pricey for me atm.
One Chance looks interesting. Thanks!

Steam only, that's a shame.

UPlay is out of question, but thanks anyway.
This War of Mine... love the game, but that's actually really depressing to play. Too much for me atm. (like Papers Please too)
Bastion? Hmm - didn't see the melancholy there but only have a played it a short time.

I think I picked that one up on Humble DRM-free a while ago. Now to find it on this garbage heap of a HDD...

Human Resource Machine looks interesting and it's already on my wishlist. Will look into it when bank account tide is better.
I've started Unrest a few days ago but kind of lacked the patience. It's still installed though.

Played that one ~20 years ago. Seems like it's only available on ... abandonware sites?

I think both games are valid choices. The music alone... actually I've installed MP1 only a few days ago.
Hmm. On the fence there. I try to avoid Humble since I keep losing track of what I've bought there...
Think I'll wait if either the German version shows up here or it's heavily discounted on GOG.

Sorry to hear about your loss. C'est la vie, and it sucks.

I remember when I first played Stalker I couldn't play too much at a time. The grey and gloomy atmosphere simply became too much for me after a while. A strong air of melancholy stays with you throughout the game, mixed with a feeling of dread and mortal fear here and there of course.
Overall an excellent FPS as well even if it is a little bit buggy in the beginning, though nowhere near as bad as some people often suggest (especially when patched)

Yeah, I've already though about re-installing STALKER. Last time I got stuck at a mission (escort...) which for the hell of it I couldn't beat (that scientist kept dying on me) and I hadn't saved to a different slot for ages....
I've kept postponing this game since I'd like to see the Tarkovsky movie first and read Roadside Picnic to be able to see how these works inspired the game.

On the whole the Eastern creators and developers really have a hand for creating an undercurrent of melancholy in their works, be it in literature, movies or games.

I just recently went through the Blackwell series of adventure games, and it's generally quite melancholy in tone. The first one is a bit clunky, but if you can bear with it and go into the rest, it's full of sad, often bittersweet stories of grief and loss. It's an adventure gaming great, IMO, and all of its moments of levity are underpinned by loss, which is something that videogame writing very rarely puts together well.

There is a working mirror link very close under the download-link. I just clicked on it and it works. Just click on the word "mirror".

Just in case that you don't see it:
http://www.adventuregamestudio.co.uk/site/games/game/1303/mirror/

Lots of good suggestions thus far. Something that I haven't seen mentioned yet is Shadowrun: Dragonfall, which captures that "bad things happen and everyone just finds a way to continue on" vibe. Transistor does something similar, too.

Ah, thanks. Was in a hurry and didn't see it.

Got both of them. Started with Shadowrun Returns recently.