Well I don't really like to throw around the "best friend" title. But here's how I met my closest friends:
INTJ: We're cousins. So we met as babies, I guess? She and I have this weird telepathy thing in large groups. We like a lot of the same things and just "get" each other. It's always been this way. Also we balance each other pretty well. Most of our time together consists of going to bookstores. She basically inhales books like candy. She'll buy an average of 5 books every time we go and her home is just walls of books. It's awesome. I like to read too, but she's one person I can't keep up with. We also get together to make dinner and discuss obscure topics/ideas while our favorite childhood movies play in the background. We've also written some stories together. I'm more into writing though, she's more into reading. It's an odd and beautiful friendship. I'm always at ease when I hang out with her. (Side note: I don't think I've met an INTJ I didn't like. Not sure if it's because INFP's get on with them really well, or if it's because they share traits with her so it feels familiar.)
ESTJ: We met through our parents about 7 years ago. My mom was the lead singer in a band. Her dad was the drummer. We started talking and she insisted we go downtown for drinks and ditch the show because she was bored. Haha. I went along with her and somehow we just clicked right away. I think she became one of my closest friends that night. We lost track of time and talked for hours. We closed down the bar (2am) and even went to a 24 hour coffee shop after to keep talking. I think we found some kind of enigma in each other. We often discuss how strange our friendship is and that it doesn't make any sense. Somehow it works. Her strengths are my weaknesses and vice-versa.
ISFJ: We met in middle school. We were both shy. We sat next to each other in band but never spoke. We started hanging out because this really loud and dramatic girl befriended both of us. We ended up at the same table at lunch and once we actually started talking we clicked. We were really close throughout high school. She actually became a very talkative person for an introvert. She's so self-sacrificing sometimes to a fault. Also, if I would say something awkward in a conversation, she had this magical way of making it not weird, even in large groups. It was a great friendship. We went to disneyland together senior year and we were in the on the dance team throughout high school so we had a lot of time to bond. We lost touch because I started dating an ENTP that she hated passionately. (Hindsight, she was right about that ENTP.) I don't see her much anymore because she's very busy with her kids, but whenever we do get back together it's like no time has passed.
INFJ: We met in high school through our dance team. She is also friends with my ISFJ friend. She was kind of the ring leader. She planned a bunch of bonding type things for us to do. She always wanted to connect with everyone on an extremely personal level. That darn Fe. lol. She does sometimes get mad if she doesn't feel appreciated due to me being late to things that she planned out for us to do. Our dynamic is pretty good though. We're on the same level but sometimes speak different languages. It can get awkward. She's a great person though. She's always volunteering for different organizations to feed the homeless or the like. We'll go to antique stores or just stay in and just talk one time, and the next she wants to go out with a huge group of people. To be honest, I get a little lost in the shuffle when we go out with her more extroverted friends. She is definitely an introvert but somehow manages to blend in depending on who she's with because she's super people oriented. To be totally honest, I haven't totally figured this one out and I think she likes it that way. She occasionally has an air superiority but I don't think she means to.
Sorry, this was really long.
TL;DR
INTJ: Met as babies. Cousins.
ESTJ: Our parents were in a band together. Met at a show.
ISFJ: Middle school cafeteria.
INFJ: Dance team auditions.