“See you later, alligator!”
“After a while, crocodile!”
 
“See you later, alligator!” is 1950s teen slang for “goodbye.”
 
“After a while, crocodile” also means “goodbye” and was often said in response. “See ya later alligator” and “in a white (sic) crocodile” was printed in the column “Teen Biz” by Suzanne Kramer in the Franklin (LA) Banner-Tribune on December 27, 1951.
 
   
(Oxford English Dictionary)
alligator, n.
b. colloq. (orig. U.S.). see you later, alligator and variants. A catchphrase used on parting.
Popularized by Bobby Charles’ 1955 song ‘See you Later, Alligator’. The expected response is ‘In (or after) a while, crocodile.’
1954   Reno (Nevada) Gaz. 16 Feb. 4/7   See you later, alligator: meaning good-bye.
1955   R. C. Guidry See you Later, Alligator (MS) 1   See you later, alligator, After ‘while, crocodile;... Don’t you know you cramp my style?
     
Newspapers.com
27 December 1951, Franklin (LA) Banner-Tribune, “Teen Biz” by Suzanne Kramer, pg. 2, col. 6:
“See ya later alligator” and “in a white (sic) crocodile.”
 
3 January 1952, Franklin (LA) Banner-Tribune, “Teen Biz” by Suzanne Kramer, pg. 5, col. 6:
See ya later alligator!!
 
7 June 1953, Seattle (WA) Times, “Seattle Teen-Agers Get Change to Test Driving Skills,” pg. 18, col. 8:
“Dig you later, alligator,” Gwen called back.
 
24 December 1953, Oxnard (CA) Press Courier, pg. 15, col. 1 ad:
Don Tosti
and his RCA-VICTOR Recording Orchestra
Composer of ... “Later, Alligator”
 
12 February 1954, The Evening Star (Washington, DC), “Teen Scene” by Sharon Doran, pg. B-2, col. 2:
Other lingo comes from Pat Dodson, a 16-year-old junior at Falls Church High. “A peon”—an underclassman; “a little later, alligator”—see ya around, kiddo; “after a while, crocodile”—I’ll still be seeing you, but sooner than the alligator one.
 
16 February 1954, Oakland (CA) Tribune, “Do You Dig This Crazy Teen-Talk?” by William Morris, pg. 15, col. 1:
See you later, alligator: meaning good-bye.
 
28 February 1954, Los Angeles (CA) Times, “Do Kids Speak English?” by Beulah Racklin, This Week magazine, pg. 28, col. 2:
“See you later, alligator.”
 
11 January 1956, Childress (TX) Index, “Adults Mystified by Slang Thrown About by Teen-Agers” by Yvonne Helmstetler, pg. 3, col. 2:
See you later Alacator. “After While Crocodile.”
 
OCLC WorldCat record
See you later, alligator
Author: Bobby Charles; Hal Leonard
Publisher: [S.l.] : Arc Music Corp., 1956.
Edition/Format:   Musical score : No Linguistic Content
 
8 February 1956, Zanesville (OH) Signal, “The Signal’s Listening Post,” pg. 4, col. 2:
“See you later, Alligator.” It’s one of those crazy new songs. Also a catch-phrase with the younger set.
 
The next line is: “After while, Crocodile.”
 
Real crazy, don’t you think?
 
OCLC WorldCat record
After a while crocodile—: the 50’s anthology
Author: Bobby Charles
Publisher: [United Kingdom] : Great Voices of the Century, 2010.
Edition/Format:   Music CD : CD audio : English