A listing of my favorite/what I consider to the be best covers of Beatles songs. I've listed them in quasi-order..."quasi" in the sense that this is highly subjective stuff.
1. In My Life
Cover by Johnny Cash.
Ponder this: John Lennon wrote "In My Life" when he was in his early 20's, but yet a man who was in his 70's and near death records it and bring a a certain level of gritty authenticity to the tune. Listen to the Cash version and you will hear the song for the very first time. It's that good.
Trivia: This is one of my daughter Rebecca's favorite songs. One of mine too.
2. Lucy In The Sky With Diamonds
Cover by Elton John & band.
I think one of the hallmarks of a great cover is that the re-recording artist brings something new to the song. That's definitely the case with this version of the Sgt. Pepper's tune. Elton's version has different sound to it, and the pacing of the vocals is completely different.
Trivia: John Lennon plays guitar on this song & also contributes backing vocals (listen carefully to the chorus and you can hear him). He is credited for the guitar work as "featuring the reggae guitars of Dr Winston O'Boogie".
Cover by Paul McCartney, Eric Clapton, et all. From the Concert for George.
I know this may not technically be a cover, since it is being played in part by another Beatle, but remember that Something wasn't a Lennon/McCartney song. I just love this version of the song, especially the initial piece where it is just McCartney, a ukulele, and some light percussion. This makes me want to sing along. As originally recorded this song was a pretty heavy love song (Frank Sinatra called it one of his favorite love songs once), but at the hands of McCartney and Clapton is is transformed into what is at first almost a sing-along...then Clapton & company (such as the terrific Andy Fairweather Low) transform it back into a serious piece of emotional angst.
Trivia: Rumor has it that McCartney didn't want to participate in the Concert for George and had to basically be shamed into joining the concert by Eric Clapton.
4. Across the Universe
Cover by Rufus Wainright.
I think this was one of John Lennon's favorite songs songs, and I can't help but think that Lennon would really like this version. It's all in the vocals by Wainright...he has this gritty but simple quality to his voice that makes the song come to life. It almost has a spoken, poetic quality to it that I think really brings the words of the song to life.
Trivia: The lyric "Jai guru deva om" roughly means "Give thanks for the heavenly teacher".
5. Got To Get You Into My Life
Cover by Earth, Wind and Fire.
McCartney wrote this as something of a tribute to Motown, which then gets turned into an actual genre piece by a soul/R&B band (Earth, Wind and Fire). This speaks to just how authentic a song-writer...and fan of music in general...McCartney actually was and is to this very day. Truth be told, as a youngster I actually heard this version of the song long before I heard the Beatles version.
Trivia: This was in the wretched St Pepper's movie spawned by the Bee Gees.
Bonus: My two favorite covers performed by the Beatles:
Money, by Barrett Strong.
Lennon just belts this one out, and the rhythm of guitars throughout the tune makes you just want to tap your feet. As is the case in many early Beatles songs you can barely hear McCartney's bass guitar.
Twist and Shout by Phil Medley and Bert Russell
Once again Lennon proves that he can belt out a rock-n-roll tune. I love the parts in the middle & end of the song where each of the Beatles contributes an "ahhh". At the very end you can hear one of them yelling "yeah". I believe that most of the early Beatles stuff like this was recorded live, meaning that all four Beatles played at the same time, instead of the instrument and vocals being recorded individually.