Vault Boy, the amazing retro mascot who appears in Pip-Boy images and Vault advertising. He's so popular, he even made it into Super Smash Bros.. as a Mii costume!
Harold, who has been in Fallout 1, 2, and 3, and gone from a human to a ghoul, and then finally to an undead un-living tree thing.
Dogmeat, a dog that can join your team early in the game, was so popular that he returned in every major Fallout game to date, with the exception of New Vegas.
Veronica has also become quite popular. This is in no small part due to her being voiced by Felicia Day and the Mojave BoS' Only Sane Man.
JoshuaGraham only shows up in the game's shortest DLC, and his bigger, active role is only by the end. And yet the amount of attention he gets is huge. Helped by the fact he's among the game's biggest badasses, what with getting thrown into the Grand Canyon while on fire and walking it off.
Sunny Smiles (and her dog, Cheyenne, by extension), found in Goodsprings, is beloved by large portions of the fanbase. This is in no small part because she's remarkably friendly, helps teach the Courier how to survive in theMojave Wasteland, rather easy on the eyes, and always encountered early-on in the game due to her being literally in the town the Courier wakes up from after getting shot in the head (creating a sense of familiarity with her). Many fans were disappointed that she wasn't a companion or even a romance option.
Easy Pete has acquired memetic status as some kind of hardcore unkillable God. It's in part because usually in evil runs he's the first one to get killed.
Magnolia, the singer at The Third Rail in Goodneighbor. She has a mysterious past, is gorgeousand a non-companion romance option, remarkably friendly, and is even voiced by Lynda Carter! She also sings her own unique songs that Lynda Carter actually wrote for the game - all of which are pretty damn good and catchy.
"The Grenade Guy", an unnamed, unseen NPC mentioned by Raiders in idle dialogue at various locations. He's some kind of delusional Michael Winslow-esque lunatic who escapes from encounters with Raiders by making sound effects, with his signature bit that earned him his nickname being to mime throwing grenades at Super Mutants, complete with the sound effects of pulling the pin and the grenade exploding. It turns out that this is Crazy Enough to Work, since it confuses and disorients the Mutants enough for him to make his escape, which he does while pretending he's on a motorcycle. Many players have expressed disappointment that it's not actually possible to meet "Grenade Guy" In-Universe.
There's also the Vault-Tec Rep, the salesman who saved the Sole Survivor's family in the prologue but himself failed to enter a Vault. Two hundred years later he's a lonely Ghoul living in Goodneighbor. After he talks about how miserable he's been since the war, most players choose to offer him a job in an outpost, which brightens his outlook considerably. The fanbase largely enjoys giving him another chance to find some honest work and make some friends after two centuries as a societal outcast.