So I have a theory about turning acquaintances into friends after empirically noticing it myself.
Essentially, you have to alternating either trading or battling with one another. So, for instance, if a Passerby asks you to trade, you accept, trade a little, and exit. Then YOU ask him to trade, if he accepts, you trade some more, and exit. I noticed that that seems to prompt the adding a friend option.
I haven't done much battling, but it just happened with me. I traded with some passerby who was collecting Vivillon forms (if you were wondering, she had mine). I offered to trade some more with her. Then after she challenged me to a battle (that I literally won with only one hit until death), the prompt to adding her as a friend appeared.
I'm wondering if you guys noticed that too. This was killing me for a while about how to get it done until like recently. It'll be hard to test this, given the nature of humans and their own motives, but it's a theory worth attempting to test.
There's an ingame notice on the PSS page that says that interacting with someone once makes them an Acquaintance, and interacting with them a second time gives you the option to make them a friend. It doesn't matter who initiates the second interaction, and it doesn't matter if it's a trade or a battle.
Well, the thing is I tried initializing the trade with someone a few times and nothing happened until I accepted something from the other person. That's what brings up this theory.
i traded with this german dude like 20 pokemon and afterwards it was like "you wanna be friends" and i was like nah
I always end up accepting everyone. Even this random Japanese person (I think it's a guy, but the console's Mii looks like a girl).
Some people I regret though like this one guy that keeps trying to trade with me every 5 minutes he's on.
ill have to test this theory with my not-yet-registered-as-a-friend friend.