Today's pairing is two original main cast Losties (well, technically one was listed as a guest star in the Pilot, but she's considered a Season 1 main cast member) who really didn't have a lot of interactions until one big episode, where they became big symbolic counterpoints to each other: Boone Carlyle and Claire Littleton. As two of the original Losties who were among the young, cute group, early speculation would have had a point in guessing that they might eventually have been written as having some sort of romantic geometry going on (Boone would have been a nice counterpoint to Charlie), but Boone was more interested in his (step) sister, and that big bald stud, John Locke. Also, Claire's Aussie accent doesn't wrap around "Boone" quite as well as "Cholly."
Boone and Claire are both on the beach following the crash of Flight 815, but Boone is too busy running around asking for pens (for Rose) to apply any of his supposed lifeguard medical emergency expertise to Claire, so the job of helping her falls to the even less capable hands of Hurley. Given his knack at bungling things, though, if Boone had been watching out for Claire in those early moments, she probably would have gotten smooshed by something, or blown up, thus making sure she'd forever stay a guest star.
The next time Boone and Claire interact a little is when she organizes the memorial for the dead passengers, along with Hurley. Soon after, Boone gave Claire one of the bottle waters he had stolen. Again, this is a case of Boone trying to help, but bungling it and making things worse. Yay Boone.
Boone's next couple of connections to Claire all involve that whole Ethan thing. He volunteers to help look for Claire when she and Charlie are abducted, and not long into the wild, Locke suggests that he and Boone split up from Jack and Kate, which started Boone's road to being Locke's little lapdog, as they stumbled upon the Hatch later that night. It was while on one of their little adventures in the jungle (looking for Vincent) that Boone and Locke came upon Claire, after she escaped from Ethan and the Staff station. What a coincidence that Vincent basically led them right to her. Or rather, that totally wasn't a coincidence, I'd say. Claire's return led the Losties to "circle the wagons", and Boone volunteered in his best intentional way, but like the goof that he was, he fell asleep, and was awoken by the trap closest to him, suggesting that Ethan probably walked right by him after killing Steve*.
And then, (nearly) finally, the story lines of Boone and Claire came together in "Do No Harm", following Boone's fall in Yemi's Beechcraft, and Locke dropping him off with Jack. The Beechcraft would also have later impact on Claire's life, when Charlie (with help from Sayid) discovered the wonders of breaking Virgin Mary statues. As Boone laid there by the caves, dying, baby Aaron decided that would be a great day to make his entrance, since it would nicely keep the Island's population balanced (not counting the dozens of people who died in the next couple of months, following). Because Boone required Jack's attention, the good doctor was not able to attend to Claire's birthing, which allowed Kate, but more hilariously, Jin and Charlie, to all have much larger roles in Aaron's birth than they probably would have otherwise. And then Boone died.
Boone and Claire were destined to have one more scene together, sort of. After the Swan station blew up, crazy old Locke decided to build a sweat lodge and ingest some of that psychedelic paste he had put on Boone's head. Boone appeared as a vision, and took Locke back to the Oceanic terminal in Sydney, and said that someone there needed Locke's help (it was Eko). The first people Locke sees are Charlie, Claire and Aaron, who seem like a perfectly happy little family. Boone says, "Not them. They'll be fine... for a while." Which, of course, was a reference to Charlie dying later in that season.
Curiously, the story may not be totally over for Boone and Claire. Like everyone who dies on the Island, Boone switched from being a Lostie to being a Ghostie. And the Ghosties seem to be connected to Jacob's cabin, and that's where Claire was last seen on the Island, and then she herself became a vision to Kate, warning her not to bring "him" back to the Island (which probably meant Aaron, but she might have meant Locke). Anyway, since they are both in that sort of ghostie state, it seems possible that maybe Boone and Claire might actually end up seeing each other, some day. If Ghosties can fall into peril, needing an incompetent hero type to bungle his way through a rescue.
*Dude, that was Scott.