Wentworth & Flexner, Dictionary of American Slang (1960) has a lengthy and interesting entry for this term, which I had mistakenly thought was connected to the year 1923:
twenty-three skid(d)oo! 23skid(d)oo! exclam[ation] A mild expression of recognition, incredulity, surprise, or pleasure, as at something remarkable or attractive; also used as an expression of rejection or refusal, sometimes as = "Go away!" "Beat it!" or "I don't care!" [Example from] 1926: "[Approx. 25 years ago] appeared in my vocabulary that effective but horrible '230Skiddoo.' Pennants and arm bands at shore resorts, parks an county fairs, bore either  or the word 'Skiddoo.' In time the numerals became synonymous with and connotative of the whole expression." C. T. Ryan, A[merican] S[peech], 2/92. Like "shoo-fly," "twenty-three skiddoo" was often used without specif. meaning. It was in male use c1900–c1910, orig. among students and sophisticated young adults. It was perh. the first truly national fad expression, and one of the most pop. fad expressions to appear in the U.S. Almost fifty years after it has had any serious use, it is still universally known and remembered. Ironically, it is now assoc. with the 1920's and is freq. used to convey the spirit of the 1920's in novels and plays of the period.
Interestingly, the third edition of Dictionary of American Slang (1995), thoroughly rewritten by Chapman & Kipfer, doesn't have an entry for "twenty-three skiddoo" at all.
Hendrickson, The Facts on File Dictionary of Word and Phrase Origins (1997) echoes the Wentworth & Flexner description of the term's meaning and historical context, and then offers this comment on its origin:
As for its derivation, it is said to have been invented or popularized by that innovative early comic-strip artist "Tad" Dorgan [who is also said to be responsible for coining hot dog and yes man, among other terms] Regarding its composition, skiddoo may be a shortening of the earlier "skedaddle." Twenty-three is a mystery. Perhaps it was a code number used by telegraphers. There is even a theory that it "owes its existence to the fact that the most gripping and thrilling word in A Tale of Two Cities is twenty-three": Sidney Carton, the 23rd man to be executed on the 23rd of the month.