For me, the main ingredient for a cover song is adding something new - preferably something unexpected, but not for the sake of novelty. When they remade Psycho in 1998 and didn't alter a single thing other than the color, audiences wondered 'what was the point'? I mean, if it's going to be the same thing, I can't see any real reason to remake something.
Personally, I think Madonna's cover of Don McLean's masterpiece, "American Pie", is the worst cover song ever. It not only brought nothing new to the song, it took away a good deal. It was like taking Moby Dick and turning it into limerick. Of course, The Brady Bunch had covered "American Pie" decades earlier and just as badly, but with less pretension.
And one more thing: I don't expect a cover version to be better than the original. Sometimes it is - not often, but it happens. I would argue that Joan Jett's version of "I Love Rock and Roll" is better than the original by The Arrows in 1975. Britney Spears' version of the same song is wretched, and far, far worse than the original. Not surprisingly, when asked about it, she attributed the song to Pat Benatar!
Anyway, I say all this to bring up the fact that Junior Parker's version of "Tomorrow Never Knows" is an example of a Beatles cover that really brings something new to the table. His album Love Ain't Nothin' but a Business Goin' On actually contains three Beatles covers, but this is my personal favorite. It's not better than the original, but it's so different, it's almost unfair to compare the two. The original is a sweeping psyechedelic journey through space and time Parker's version is a dark and somber look into the void. A rare occasion where a Beatles cover is arguably as brilliant as the original. Suffice it to say, Parker had come a long way from "Annie Get Your Yo-Yo".
Fairly recently, the song was used in the soundtrack to Children of Men, which also contained the King Crimson classic, "Court of the Crimson King". The film wasn't too bad, but this is surely another case of a soundtrack being better than the movie (a la Platoon and I Am Sam)..... I'm rambling - here's the song.