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.