Not sure exactly what the intent of the question is, but my favorite Beatles covers are "Twist and Shout" and "Roll Over Beethoven." "Rock and Roll Music" is a close third. Maybe "Dizzy Miss Lizzy" after that. I've always had a great liking for "A Taste of Honey", too, but I know a lot of people don't seem to care much for it.
The absolute last place would be "Til There Was You." That was an odd choice for the Beatles to cover, it being a standard from musical theater and a little bit sappy.
You can see a lot of what influenced the Beatles in their cover choices. They were huge fans of Little Richard, Carl Perkins, Buddy Holly, among others. Paul's singing, especially on the fast numbers, is very reminiscent of Little Richard's.
The Beatles name, of course, was inspired by Buddy Holly & the Crickets. Buddy liked the Crickets as a name because of the harmonious night songs of the little insects. In England, "cricket" has other meanings...there's the game, cricket, but also when something is "cricket", it's meant to be straight up and good. So John loved the name and came up with beetles spelled with an "a" to imply musical beat. An earlier name of the group was "Long John and the Silver Beatles". That was also some clever Lennon wordplay, invoking Long John Silver of "Treasure Island."
John was asked early on how they chose the name "Beatles." He wrote that a man on a flaming pie appeared and said that henceforth they would be Beatles with an 'a'.