The first guess might of course be that a newspaper announced to stop circulation, perhaps only of its 'morning edition', or the entire outfit is about to close down. Thus final edition, ever.
The Evening Sun publishes its final editions today,…
A more likely explanation however is that compared to a bulldog edition, the final edition of a daily or morning paper is really the latest edition of that version for the day:
Afternoon or evening papers, once common but now scarce, are aimed more at commuters and office workers. In practice (though this may vary according to country), a morning newspaper is available in early editions from before midnight on the night before its cover date, further editions being printed and distributed during the night. The later editions can include breaking news which was first revealed that day, after the morning edition was already printed. Previews of tomorrow's newspapers are often a feature of late night news programs, such as Newsnight in the United Kingdom.
Like explained in this article:
The latest evening paper to try a morning edition is The Evening Bulletin in Philadelphia. It formerly came out around 8 A.M., but an early edition now goes to press in the middle of the night and hits key downtown newsstands by 6:30 or 7 A.M. and somewhat later in the outlying areas.
Meaning the content from the first to the final run can change in any unpredictable way, updated is what we would call that today.
That is in contrast to what is explained in the related stop-press article, as that describes an unforeseen and unintended change of the production run, whereas with a 'final edition' these breaks and changes are intentional.
An example for this practice can be illustrated with this final edition from December 7, 1941, which was quite fast, but still differs markedly from what most papers wrote on December 8:
— Dec 1941 Newspaper Declare War On Japan Final Edition Gary Post Tribune
The show is Alfred Hitchcock Presents, Season 3, Episode 17, (95) "The Motive"..
As such, this is another ephemeral non-Chekhov-gun Hitchcock uses. Not much clues from the episode itself. Except this bit:
In the show watched the dialogue in context reads
The morning paper, sir.
Is that the final edition? Yes, sir.
You going somewhere, Tommy? Have been.
Yes, my dear boy, I have been.
You're being pretty cagey with an old pal.
Well, you can read all about it in the morning paper.
"Jerome Stanton, chemical engineer, "was found murdered in his home last night.
"The police have already staked out a suspect on a clue furnished by Mrs.
"When asked if she knew who might have had a motive for the crime, "she replied that her former husband, Tommy Greer, "
So it's likely a very current very local event that happened during the night when first edition might/will be already be printed without having that news, necessitating a later run of the regular morning edition to be able to read about it in print.