He also, and perhaps more famously, covered “You Can’t Hurry Love” (originally by The Supremes in 1966).
Here are the ones that immediately sprang to my mind, which I’m sure are only a tiny sample of songs that fit this description:
“Crystal Blue Persuasion” covered by John Wesley Harding (original by Tommy James and the Shondells, 1969)
“Time of the Season” covered by Trip Shakespeare (original by The Zombies, 1968)
“Can’t Take My Eyes Off Of You” covered by Daniel Amos (original by Frankie Valli, 1967)
“California Girls” covered by David Lee Roth (original by The Beach Boys, 1965)
Wendy by The Beach Boys ( 1964 ) and by The Descendants ( 1986 ).
Thank you everyone or the replies 
Little Eva did the original version of Carole King’ s “The Loco-Motion” (Eva was Carole’s babysitter), but it was a #1 hit for Grand Funk in the Seventies and a #3 hit for Kylie Minogue.
The Supremes had a smash with “You Keep Me Hanging On,” which was later a hit for both Vanilla Fudge and Kim Wilde.
The Marvelettes’ early Sixties hit “Please Mr. Postman” was remade by the Carpenters in the Seventies, and went to #1.
In my high school days, Shaun Cassidy hit #1 with a remake of the Crystals “Da Doo Ron Ron.”
Anne Murray had a Seventies hit with a remake of the Monkees’ “Daydream Believer.”
Several of Roy Orbison’s early Sixties hits were later re-made successfully. Don MacLean had a hit with "“Crying,” "Linda Ronstadt had a hit with “Blue Bayou,” and Van Halen had a hit with “Pretty Woman.”
Nazareth’s smash hit “Love Hurts” was a cover of an Everly Brothers song from the Fifties.
Back in the Eighties, two consecutive #1 singles were covers of Tommy James & the Shondells. You already mentioned Tiffany’s “I Think We’re Alone Now,” but there was also Billy Idol’s “Mony Mony.”
Tiffany also had a hit remake of the Beatles’ “I Saw Her Standing There.”
Joan Jett had a top ten remake of Tommy James’ “Crimson and Clover.”
Paul Simon did a great cover of Surfer Girl at a Brian Wilson tribute show.
“Just a Gigolo”/“I Ain’t Got Nobody”: Louis Prima in '56 (though used in his Vegas shows earlier and the individual songs date back to the '20s & '10s, respectively), David Lee Roth in '85.
Amii Stewart had a #1 hit with her disco version of Eddie Floyd’s Sixties R & B song “Knock on Wood.”
Michael Bolton had several big hits with covers of Sixties songs: Otis Redding’ “Dock of the Bay,” "Ray Charles’ “Georgia on my Mind,” and Percy Sledge’s “When a Man Loves a Woman.”
Los Lobos had a #1 hit with a remake of Ritchie Valens’ “La Bamba.”
UB40 had hits with covers of Neil Diamond’s “Red Red WIne” and Elvis Presley’s “Can’t Help Falling in Love With You.”
Cheap Trick covered ELvis’ “Don’t Be Cruel” and Fats Domino’s “Ain’t That a Shame.”
Devo covered the Stones’ “Satisfaction.”
Hall & Oates remade the Righteous Brothers’ “You’ve Lost That Loving Feeling.”
George Harrison’s biggest hit, “Got My Mind Set on You” was a cover of an early Sixties soul song by James Ray.
In addition to Da Doo Ron Ron (1963), Cassidy’s 1976 debut album also had covers of Take Good Care of My Baby (orig. Bobby Vee, 1961) and Be My Baby (orig. The Ronettes, 1963).
Incidentally, the second single off Cassidy’s debut, That’s Rock ‘n’ Roll, was written by Eric Carmen, the lead singer of the Raspberries (Go All The Way) and who had later solo hits with All By Myself and Hungry Eyes.
(Sean Cassidy was my second album, right after Bay City Rollers Greatest Hits. I was seven.)
As Biggirl noted, Linda Ronstadt had big hits with covers of songs by…
- The Everly Brothers (“When Will I Be Loved”)
- Chuck Berry (“Back in the USA”)
- Buddy Holly (“That’ll Be the Day” and “It’s So Easy”)
- Smoky Robinson & the Miracles (“Tracks of My Tears” and “Ooh Baby Baby”))
- Martha & the Vandellas (“Heat Wave”)
Mick Jagger & David Bowie had a hit remake of Martha & The Vandellas’ “Dancing in the Streets.”
Robert John had a hit with a remake of the Tokens’ “The Lion Sleeps Tonight.”
The Captain & Tennille remade Smoky Robinson’s “Shop Around.”
The Spinners had a hit remake of Frankie Valli’s “Working My Way Back to You.”
There have been several hit remakes of Fifties and early Sixties hits by Sam Cooke:
- Dr. Hook’s “Only Sixteen”
- Both Tony Orlando and the Spinners both had hit remakes of “Cupid.”
- Rod Stewart covered “Twisting the Night Away.”
- Art Garfunkel did a hit remake of “Wonderful World” with James Taylor and Paul Simon
- Cat Stevens had a hit remake of “Another Saturday Night.”
Robert Plant and the Honeydrippers reached #3 with their remake of Phil Phillips’ oldie “Sea of Love.”