No one, including George himself, really seems to like "Don't Bother Me" that much, but I've always been a "Don't Bother Me" apologist. It's actually got a lot of interesting stuff going on in it, especially given that it's a first song, that it appears on such an early Beatles album, and that it's so different from the Lennon-McCartney songs that had so far made up the bulk of the Beatles' work.
For one thing, the song is largely in the Dorian mode, which might seem like a pedantic and meaningless thing to point out. But I always like to think of it, because this song was one of the first times that I was able to apply what I was learning in theory classes in college to my life-- I actually remember being in my freshman dorm room and "Don't Bother Me" coming on, soon after we'd been introduced to the medieval modal system, and all of a sudden I was like "this sounds Dorian!", and I played it on the piano and it totally was. So I guess all my geeking out on Beatles songs might have begun right here with "Don't Bother Me," because after that I tried hearing stuff like that everywhere. (And if you're listening for modal material in Beatles songs, you won't be disappointed-- they particularly like to use a pinch of Mixolydian here and there.)
Anyway, when I say Dorian mode, what I'm really talking about is that it's basically in a minor key, but one shade brighter than a traditional minor key, if that makes sense. And even though George (and all the Beatles) surely didn't know what the modes were when they were writing, Dorian is found in all kinds of music, particularly English music (the textbook example is "Greensleeves"), and surely is something they would have been familiar with on a subconscious level. Actually, what I think is cool is that from the beginning, George and John and Paul had brains that were looking beyond the rock music they loved and finding inspiration in all the music they'd ever heard-- and all the back in 1963 with With the Beatles, way, way pre-Sgt. Pepper and its ilk, you know? This is why they are awesome.
But anyway, there's a lot more here that makes "Don't Bother Me" rock. I've read other commentators say that they hear something almost Latin in this song, which I can relate to-- you can hear Paul playing the claves, which seems a bit salsa-esque, and Ringo's got some bongo stuff going on too, all of which combines to make the percussion sound pretty thick and interesting. And if you listen to George's guitar line underneath his vocal, it's really pretty melodic, which is just contributing to the complexity of the texture.
George wrote this song while in bed recovering from an illness, so I presume the "Don't Bother Me" was a very real ornery feeling that he transformed into a sorta-kinda love song. He has said that he mainly did it to see if he could write a song, and indeed he didn't get into songwriting too seriously until later-- in fact, the world wouldn't hear another Harrison original until Help!, three albums off. But "Don't Bother Me" would have gotten me through those times if I were a teenage George fan, I think. It might be underappreciated, but it totally rocks.
See?! Look how much the people in the club in A Hard Day's Night are enjoying it! Word.
"Don't Bother Me," released in the U.K. side A track 4 of With the Beatles, November 21, 1963; in the U.S. side B track 1 of Meet the Beatles, January 20, 1964.