There are two types of lessons a child can learn from trying something and failing: either to keep trying it until he gets it; or, that it's actually okay to fail. That is, despite what his parents/teachers have been telling him, the world didn't end just because he gave up/didn't finish/flunked a course. Or, nothing really bad happened when he failed, trying hard is frustrating, so why bother trying so hard next time?
If a child fails at something he really wants, over and over, he may learn he's not good enough to have what he wants, and lower his expectations accordingly. And he may also learn that he can't trust his parents when they tell him he can do it, because they're always wrong.
It sounds like your son's had a lot of experience with giving up, to the point where that's become his go-to strategy. You want your son to learn to keep trying, but you can't learn to persist until you've seen persistence work for you. To make that happen for your son, you not only have to teach him that he can succeed if he persists, but you also have to get him to abandon something -- giving up -- that he has become comfortable with.
The way to do that is not to keep challenging him with something complicated until he pulls himself up by his boot straps and figures it out. Yes, it worked for his cousins, but people are different; your (ADHD?) son doesn't have the same type of brain as his cousins, so you shouldn't have the same expectations. (I am sure that showing a kid people swimming in a swimming pool and then throwing the kid into the swimming pool until he learns to swim would work for some large fraction of kids; but others would drown.)
Instead, I'd give him little tiny one-step challenges he can master. E.g., build him a small house in Minecraft, so he has somewhere to stay safe at night, and then challenge him to do a bunch of easy things: Mine wood. Plant a sapling. Make the house bigger. Make a chest.
When he's mastered all these individual tasks, I'd challenge him to slightly more complicated things (make a garden with a fence? make a furnace? make armor?), etc., but first I'd try to get him comfortable with his new, persistent self by pointing out all his recent successes and telling him it's an outgrowth of his old self. Tell him that people's brains get stronger and smarter when they get older. Just like when he was three, he couldn't add 4 plus 5, but now he can; and just like of course he can't change a car tire himself right now, but he'll be able to when he's eighteen... he's gotten better at persisting because his brain is getting stronger. And the more practice he gets with difficult things, the smarter and stronger his brain will get.
For even more complicated tasks (bookcases?), I would ask him to write down all the steps he's going to need. Tell him you'll check his list when he's done, before he starts the task. That way he can feels safe when he's extending himself to something tougher, and won't have to worry about failing. This works even better outside Minecraft. I get my daughter to do quite complicated things when I get her to write down all the steps first and check them for her. And she gets to take the credit, since it was her list (even though I helped her check it), and she did it.
Also try to make sure, at least for now, that he never gets put in positions where he's likely to fail. He needs to build up confidence in succeeding, for a while.
Additionally, try to show him that failure now doesn't mean giving up on a problem forever. E.g., sometimes sleeping on a problem means you find the answer easily the next day. Point out, when they happen, any situations (a leaky faucet?) where you have to give up momentarily and call in help from an expert. Point out that sometimes there is more than one way to solve a problem.
Finally, I'd suggest talking to him about problems you may have encountered and ask his advice. Get him used to being a problem solver. See if he has any neat ideas, and if so, try to implement them. E.g., "Your sister wants me to put up this picture but there isn't any room for it in the living room." He might suggest putting it up in her bedroom, and that would be something you could do then and there. Ask if he'll help. Or say, "I wanted us to watch this movie this weekend but it's too late now, it's time for bed." Hopefully he'll suggest doing it Saturday night, and you can say, "Great, good idea, we'll do it!"