Okay, let's throw a couple suggestions.

Starbound. It's a sandbox and so playing it together with someone is quite inconsequential, but it can be fun nevertheless. More fun than playing it alone perhaps even.

Classic Doom (That's DOOM II + Final DOOM as well as The Ultimate DOOM), if you're willing to use some source port. The multiplayer implementation in the original release is quite baroque.

If you can get System Shock 2 co-op working, it's an amazing game. But it might or might not work. And when it works, it is still glitchy. Beware. It does require some tactics though, as you'll be sharing the resources you find. So I don't know if I can recommend co-op for a first playthrough.

Terraria, it's much like Starbound and has well working multiplayer. Though it's not discounted right now.

Oh, and try to avoid Judge Dredd. It is shit.

Serious Sam, both encounters. It's a crapton of fun if you like simple and simply dead fucking serious first person shooters. It is especially fun in co-op. Highly recommended.

Ghost Recon, if you like actually serious shooters. You die easily, enemies die easily. Co-op is somewhat limited in that you do not get any AI teammates (you get a full squad in single player). However, it is beatable. I can think of one (maybe two) mission that is *difficult* with only two players. But you can beat it. Even today, I think GR is one of the better tactical shooters out there. The genre is pretty much dead, what with the mainstream focusing on flashy hollywood style power fantasy shooters. It's a real shame the GR expansions aren't here on GOG...

Hidden & Dangerous 2. I have not played it in co-op, and I've only played the first mission of single player. However, I lured an acquaintance into playing it, and he played the co-op with his brother. He thinks highly of it. It's a little clunky for a shooter, trying to do realism in a way that isn't super smooth. That said, if you give it an hour or two, you ought to get used to the controls & weird motion.

The Baldur's Gate series. Great RPGs but again it might be a little challenging for a first time playthrough.

Neverwinter Nights. If Baldur's Gate is good for you. NWN should be too. Actually I haven't played this in co-op. But it ought to work. NWN is somewhat focused on custom content & online play.

Torchlight 2. Action RPG cast very much in the mould that Diablo 2 created. I've played this online with my sister. Decent, if a little repetitive.

Titan Quest. Another action RPG. I've only played the beginning, but that was more promising than Torchlight 2. Some people on twitch.tv/gogcom/ praised this a lot, and there was a stream (headed by Dizzy IIRC) where he played with other visitors on the channel.

Unreal Gold. Old shooter with lush atmosphere and lovely music. I don't think it's super well optimized for co-op, with all the lifts and other interactive level elements that are prone to leaving the other player behind. It should work though. I haven't tested.

Trine (2?), if you care for platformers. I think the Enhanced Edition of the first game might also have online co-op. I have no idea about Trine 3. I know my sister played one or both of the first two Trines online with her friend, though that was the Steam version. I've only played local co-op.

Actually, I can't guarantee they all work online without hassle :( Sorry about that. But I know they all work at least in LAN. Terraria works online for sure. So if the games seem interesting in the slightest, do a bit of research. It shoud be less work than browsing the entire catalog of multiplayer/co-op games and trying to figure out what's what.

The need for hamachi and such also depends on whether you (or your friends) are behind a draconian firewall or NAT, i.e. not properly online at all :(