In a classic moment—referenced on Community among other shows—Bender complains about Claire's diamond earrings, saying they were probably a gift from her dad. He contrasts them with the harsh reality of Bender-family Christmas presents:

BENDER: You know what I got for Christmas this year? It was a banner f***in' year at the old Bender family! I got a carton of cigarettes. The old man grabbed me and said "Hey! Smoke up Johnny!"

This reveals a few things: Bender's dad doesn't care about the health risks associated with smoking, and he doesn't want to buy his kid a nice, normal Christmas present. So, it highlights the dysfunction of Bender's home…

However… An article from the AVClub on horrible Christmas presents from TV and movies makes a good point. It might be harsh and unfair toward John Hughes and Judd Nelson personally, but it does say some perceptive things about the carton of cigarettes itself:

The scene in John Hughes' angst-rific The Breakfast Club where wildly overacting quasi-hoodlum Judd Nelson mocks the diamond earrings sported by pouty rich girl Molly Ringwald is supposed to be a dramatic high point. Instead, it shows how out of it Hughes really was: When Nelson rails about how his big Christmas gift was a carton of cigarettes, his dad's "Hey, smoke up, Johnny!" comes across as more gregarious than abusive. Hughes' childhood in the tony Northbrook suburb of Chicago probably left him ill-prepared for the notion that some kids would be perfectly happy to get a carton of cigarettes for Christmas, as opposed to, say, nothing. And hey, it isn't like Nelson's old man only bought him a pack… (Source)

Still, regardless of whether Hughes knew what he was talking about, the carton of cigarettes is meant to expose how Bender's father doesn't really care about him. Even though Bender probably loves cigarettes, maybe, on a deeper level, he doesn't want his parents to just cater to his delinquency? He wants them to actually care, the way parents are supposed to, which is why he finds the carton of cigarettes such a lousy and disappointing gift. He secretly wants the sort of idyllic Mr. Rogers-as-dad home life that he thinks Brian has.