Butters and Pip were two of the three "Melvins" in the episode of that name, so obviously they were buddies at some point. Here's my take on how they met.
"Sorry, Butters, looks like we're out of room," Kyle said, and even though it was plainly obvious they could make room by moving all their books Butters accepted this and moved on with a small sigh.
"I guess I'll see you fellas later, then." He was used to this by now, every lunch period the table would "suddenly fill up" the second he got close enough to sit down. Butters didn't mind too much, it wasn't his place to demand they make room for him, but he still couldn't help feeling a little rejected.
So as usual, he went to find an empty table. Eating alone wasn't much fun, but at least he would get to sit down before a bunch of other kids grabbed the table and "suddenly" ran out of room for him. Unfortunately, there were no empty tables that day.
"Um...excuse me," he stammered to the lone blond boy sitting at the nearest table. "I-is it okay if I sit down? I won't get sore if you say no, but it's just that all the other tables are kinda full and, well, my feet are getting kinda tired." The boy looked up and smiled cheerfully.
"Oh, not at all! Please, do have a seat, I was feeling dreadfully lonely until you arrived," he said. He spoke with a British accent and dressed more like a character from those movies about the nineteen-hundreds they'd seen in class every now and then.
"Gee, thanks!" Butters eagerly took the seat across from him. "Boy, every time I saw a bench with a free spot someone suddenly filled it, I was sure getting discouraged! Um, my name's Butters. My real name's Leopold but everyone calls me Butters cause...well, I dunno why!"
"How lovely to make your acquaintance! My name is Philip Pirrup, but the other children call me Pip," the boy replied.
"Nice to meet you too, Pip! Say, now that I think about it I see you eating alone all the time, why's that?"
"Because the other children hate me," Pip said blithely.
"Oh, that's so sad." Butters frowned sympathetically, but the British boy shrugged it off.
"I'm used to it," he said. "Why is someone so charming as yourself all alone?" Now Butters was the one to shrug.
"Same reason!" For some reason this made both both boys laugh heartily. "Gee, you really think I'm charming? Well, I think you're charming too, Pip!"
"Say, Butters...why don't you and I be friends? I think we'd get along smashingly the more we got to know each other!" Butter's heart felt light, he'd never had a real friend before.
"Well, sure! I'd like that!"
Pip smiled, reaching his hand across the table to shake Butters's.
"Here's to never being lonely at lunchtime again!"