To upgrade mercs, you need to finish their Tasks.
But you can’t just choose any Task to work on; you have to wait for that specific merc to appear as a “visitor” at your campfire. Right?
Or am I wrong? If I have a specific merc that I want to improve, is there a way to ensure that they visit my campfire?
Daily mystery visitors (4) came in randomly. Nowadays their quest is chained, meaning you finish their quest 1, it goes to quest 2 and keeps on going until final task 18, then you are done with this Merc.
So my best option is to keep as many open spots at my campfire as possible, and hope that the merc I want comes to visit?
If you want to level up a specific Merc. Try following step
Mercenaries appear at your campfire at random up to 4 times. 1 spot if for a story line if something new appears and the other is for Toki or a limited quest line like C’thun. So you never know who you will get and you will have to complete all that merc’s quests or abandon it to get another character unless if all merc’s quests are completed.
Any source or confirmation for that?
As far as I know, it’s the level of the bounty that matters.
This is not sufficient for the most effective fishing for mysterious strangers. Ideally, the other members of the party that you use to get through the map should have their tasks competed or active, so that, if you get lucky and find the mysterious stranger, you are guaranteed to get the task for the mercenary you want.
For example, if you rely on Cariel, Xyrella and Samuro to get through bounties, you might wish to get their tasks done first, then carry another mercenary whose tasks you want.
PS The chances for the mysterious stranger are lower than before, so it might be a frsutrating experience — fishing for that node.
It was tested and analyzed by streamers, the more fights you had before mysterious boom the more chance for strangers to occur. They did a bunch tests to backup their theory.
As others have discussed, you can try for a Mysterious Stranger while you have the merc you want in your party. It’s very slow now, but you’ll find one eventually. If the merc you want is a useful member of a good team, it’s easy enough to just play that team and check the mystery boon in every bounty just in case. Ideally with an alternate path you can take if the “boon” is the Spud ME. 
Another method I used shortly after the current task system was implemented, when I still had a lot of mercs needing tasks, was to do tasks for the random mercs on my task board for a few bounties, then when I stop playing for the day (often when the least desirable mercs reached a slower task, like 7, 13, or 15, or one of the oddball ultra-grindy tasks some mercs have) I’d abandon one or more mercs to cycle the task board in the hope of getting a more useful merc onto it the next day. Repeat as needed until you get all the important mercs done and/or the randos progress far enough that you may as well finish off their tasks to remove them from the pool.
You’ll have to do better than that… So far sounds like a spurious and unconfirmed rumour.
Oh, yeah, planning a route for me involves taking a path that can go through a mystery node with an alternative (sometimes I might even wish to avoid Cursed Treasure if the party is more weak and the boss fight might be risky, although generally it would be highly desirable… And I don’t like Sabotage much either, sometimes it can mess up things) and avoiding those stupid ‘boons’. Elite fights are usually much less of an issue. As for a good team, it should be able to carry at least one ‘extra’ merc, if nothing else.
I watched streamer with their test data analysis and I choose to believe it. You don’t have to.
Anyone can claim anything. One has got to present actual data (a reference will do) for any meaningful discussion.
You are certainly welcome to continue to believe that, but you should be able to find several links to various tests in this forum if you really want to learn.
That’s actually confirmed in the wiki and other sources.
Again, empty words. Data? References? At least a single link, if nothing else? Nope… And thus — nothing to discuss.
https://www.reddit.com/r/hearthstone/comments/qmmb0x/hc_banehollow_mysterious_stranger_worth_it_to_do/
Rarran reported this in a stream back when he was doing Mercs.
The reddit post showed a stronger correlation for “number of fights” than for “node number”, which would argue that skipping fights to reach the node is a bad idea.
Ten months old, and blizzard has reported changes in Stranger rates since then.
I’m not aware of any reported data since the last patch. People who really care should collect and report their results.
Finally! Some actual data… 
This is an interesting exercise. Let’s take the first part of the table and, assuming binomial distribution of the chance to get the stranger (we don’t care much for anything else here), make some crude estimates of the probability to get it plus-minus the standard deviation times a certain coefficient (anyone has a quick tool to calculate these for binomial distribution? I doubt that the normal-distribution estimate, where a should be about 2 for a 95% confidence interval, is suitable here, especially for the smaller sample sizes), which depends on the confidence interval. For each of the five columns respectively, we obtain:
0.20 ± a*0.07, 0.40 ± a*0.03, 0.55 ± a*0.03, 0.66 ± a*0.04, 0.73 ± a*0.07
Now, it looks interesting at first, but even within the ‘three sigma’ (a=3) range the probabilities are more or less indistinguishable… And in particle physics, for example, where sample sizes tends to be much greater and the distribution so much closer to normal, they tend to prefer even more, such as 4-5 sigma if memory serves, to claim something definitively… And then again, I’d also appreciate if someone could actually estimate the appropriate confidence intervals for a binomial distribution like here.
‘Who in the Light’s name’, as Cariel would put it, is ‘Rarran’? Anyone can broadcast anything on the platform of their choice nowadays, more or less… I wish people would read books, use their heads, do the research and so on, whatever is appropriate for the situation at hand, rather than give in to their flock instincts and follow The-Light-knows-whom blindly, generally speaking.
I don’t know if the reporter of this data (respect for that, I’m by no means trying to denigrate anyone or their contribution) even knows what a correlation is… But I don’t feel like doing this kind of analysis here, sorry — besides, sorting out that matter of a reasonable value for a first would probably be more beneficial from a practical viewpoint.
I would assume that, in order to minimise their workload, the developers would probably adjust the rates (and add another possible events for level-30 Heroic bounties), rather than rewrite the entire algorithm… But then again, it is only a guess.
Streamers have different rng then regular players because of their (easy) free advertisement for the game. I am sure that populair streamers have some mark on them to get better loot. So others try the same but get disappointed.
Bro are you being serious? lol