These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
An unnamed (gasp!) boy appeared in one scene of manga Vol. 6. Apparently he was VP of Miyuki's class and wanted to confess to her but never did. Unsurprisingly, some fanart of him and Miyuki has sprung up. He's been given the name Toshiaki, which is basically the 2Chan version of "Anon".
Fridge Horror: Konata looks just like her mom...who died early of some unspecified condition. Some fans have thought that Konata's lazy, seemingly carefree approach to life reflects her beliefs about a shortened life expectancy.
"Funny Aneurysm" Moment: Kanata didn't want her daughter to have Sou's personality. When she died, Konata didn't have a strong female role model to keep her grounded.
Genius Bonus: Lampshaded heavily with all the anime, manga, and otaku references Konata makes. Her friends often have no clue what she's talking about.
Growing the Beard: Many see episode 5, when the first director, Yutaka Yamamoto, was terminated and replaced by Yasuhiro Takemoto, as the turning point in which the series really started to get funny.
Les Yay (Some people see this between Konata and Kagami. Hiyori often sees this between Yutaka and Minami only to chastise herself by saying to herself that she shouldn't think about her friends that way.)
"I want us to join our bodies!" Hard to get more blatant than that.
In one of the fourth wall breaking strips published in Kadokawa Hot Line catalog in 2005, Konata states that in order for Lucky Star to have a chance of being turned into an anime in the future, they need more romance in the stories. Kagami then points out the lack of men around, to which Konata responds that she didn't mention anything about men...
Strongest example is Yutaka and Minami since they have a very close relationship. Kagami and Konata are very iffy and are bi at best. Kagami does seem to be conscious of Konata at times and Konata does seem to have a thing for moe female characters though.
The anime focuses on this compared to the manga, and this is the root of all the fanart of such nature.
Memetic Mutation: To a lesser extent than most, but for most of the fans all you have to do is mention chocolate cornets...
Moe: Every main character that isn't Soujiro can be seen as Moe by someone (probably a result of Puni Plush) Konata outright designates Miyuki and Yutaka as walking Moe. And let's not forget that Kagami even won the 2008 Saimoe Tournament.
Pandering to the Base: This series is commonly accused to be this. This may be the result of narrow focus; Comptiq after all is a video game/manga magazine and comic strips on that kind of magazine is supposed to be like that...
The anime is accused of this more with all the Les Yay and Shout Outs (even compared to the manga)
Sci-Fi Ghetto: A variation is discussed in the second episode, when Konata brings up a poll to determine how many books the average Japanese citizen reads in a year - the poll specifically excludes manga, though no such restriction is placed on the comparable media of light novels and children's picture books. She goes on to wonder how people can say that illiteracy is increasingly common among young people, pointing out that blogs and web pages in general take up a major part of the average teen's free time, and they are as wordy, if not more.
For Tsukasa fans, the volleyball scene in the OVA. Her sister smacks her in the face with the ball, she tries to give it her all for once, but she still loses the game. One could reasonably argue that it wouldn't really be Tsukasa if she won, but still. She says, "Next time...", but given the "grand finale" feel to the OVA, it doesn't seem like there will ever be a next time.
The Woobie: Poor Tsukasa, whom the directors of the show REALLY get a kick out of portraying as a helpless little girl who is LITERALLY pushed around in a big man's world. Even though this does make her even more adorable.
Kagami also counts to a lesser extent. She who has to put up with constant abuse from Konata and never gets into the class she wants, not to mention her adorable Cry Cute moments.
Also when dere-dere Kagami expresses her insecurities or wants that she can't bring herself to show. Yet again for Kagami during the episode where she has to put up with her (also Tsundere) big sister. She seemed at the verge of tears at one point. *runs to hug*
Also Minami in the manga; she's often mistaken to be the one behind many no-nos, such as not cleaning up after her dog, or a prank involving pressing all the buttons in an elevator.