Searching Google for the history of the slang term "the bomb" (as in "That song is the bomb") yields a number of results in 40s/50s jazz glossaries, but they tend to at best give an artificial example of usage, nothing about the actual history behind the usage. Here's one such glossary (there are numerous others): Jazz Slang.
I'm wondering if this is based on some actual history of the term, or if a bunch of "jazz glossaries" are just copying from each other, especially since there are some other comprehensive glossaries that do not include the term. I also very much thought the term was more modern than the Jazz Era, but I could be falling for the recency fallacy there. Then again, considering that the atomic bomb was dropped in the 40's it's feasible that the term originated in the Jazz scene.
The oldest reference I could find of this usage is from 1973 here, but I don't have access to the OED, so I'm not sure if there's an older reference in there. Several regular dictionaries include this meaning, but either don't list any origin or only say "Origin: US".
I found an article on dictionary.com, Lexical Investigations: Bomb, which mostly focuses on a different related terms (just "bomb" not "the bomb"), but in passing says "the bomb" originated in Great Britain and rubbed off on Americans in the 90's. But it doesn't give any citations or references to support that.
Related question about the term "cool", which did originate in the Jazz scene: Where did the slang usages of "cool" come from?
Best Answer
Summary
I can't find any evidence to suggest this is a jazz term, and the earliest example I found is from 1995. The alternative da bomb dates from at least 1994.
There's no entry in a number of slang dictionaries (A Jazz Lexicon, A Dictionary of Slang and Unconventional English, Shorter Slang Dictionary, The Slang and Jargon of Drugs and Drink, A Dictionary of Cliches, American Heritage Dictionary of Idioms, British English A to Zed). This suggests it's a recent term.
'50s claims
Like your linked Jazz Slang, it's often claimed to be 1950s or jazz slang but there's no evidence for it. This list of 50's Slang includes:
The list is undated but was first archived in 2002. Further, The Crusaders' released Louisiana Hot Sauce in 1996. The others all seem to be copies of this (or another).
OED
Although not exactly the same, the closest thing in the OED is:
Here's the first and a couple of other quotations:
All the quotations are "a bomb" not "the bomb" and are quite different.
Lexical Investigations
The only relevant quotation from your linked Lexical Investigations: Bomb is fairly recent:
Google Groups
The earliest "is the bomb" I found in Google Groups (there may be earlier, but Google recently redesigned Groups and crippled the search) is this exchange from August 1996:
There's nothing before 1996, and quite a few after and into the 2000s and this decade.
Subzin movie subtitles
The earliest I found in film subtitles is the 1995 Spike Lee film Clockers which uses it twice:
And:
The film was released in September 1995 and based on a 1992 book by Richard Price, although the book doesn't appear to use the phrase.
"Da bomb"
There's even more and earlier Google Groups results for the variant da bomb, such as this from soc.culture.filipino in December 1994 ("SCF at REDj is DA BOMB...."):
A few days later in the same group ("*** DA BOMB ***"):
There's an earlier use in comp.sys.mac.apps from July 1994. The post discusses Apple computer crashes ("Has anyone else had problems with Appleshare causing a system bomb on startup?") but the subject seems to be a passing reference to the phrase: