History of "have a good one"

This is, as you noted, a very American idiomatic phrase which apparently dates from the 1970s. One dictionary includes the phrase in its definition of good, but notes it is American English. An idiom dictionary further refers to the phrase as a cliche. Interestingly, the only idiomatic definition they give is from the McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

One should be cautious when using this phrase, however. A blog found that the British explanation of the origins of the phrase were:

Have a nice day

Meaning: A salutation, ostensibly to offer good wishes. In fact a banal and insincere form of words given to anyone and everyone. Evidence of the meaninglessness of the sentiment is the fact that it is even used last thing at night when the opportunity to have a nice day has all but disappeared.

Origin: US origin - around 1970s.

The blogger notes that British people may construe this parting phrase as highly sarcastic, so it may be best to keep its use to the American English arena.

Having grown up with that phrase in Colorado, I know how it was used there: In the context of having a good run (down the ski slope). For a long time, it seemed unique to Colorado. Although I lived elsewhere, I would only hear it used (with pleasure) when I returned to visit Colorado, although even there it was being used more and more often outside of the original common usage. I read online it comes from military history relating to paratroopers heading out of the plane and that having a good "one" meant landing.

It's not appropriate, in my opinion, to use the idiom related to everyday affairs such as leaving the grocery check-out counter, which makes it equivalent to "have a nice day." It belongs to special events: a ski run, a landing, a race, a game, a date, etc., where the "one" is specific.

The Concise New Partridge Dictionary of Slang (2007) says:

have a good one goodbye. Slightly cooler than urging someone to 'have a good day' US, 1984

The American Heritage Dictionary of Idioms (1997) says:

have a nice day Also, have a good day; have a good one. A cordial goodbye ... For example, Thanks for the order, have a nice day, or See you next week — have a good day, or The car's ready for you — have a good one. These expressions have become synonymous with a polite farewll. The first originated about 1920 but, like the variants, became widespread only after 1950.

The Grammarphobia Blog says:

The earliest appearance is from the headline of an article about Washington’s Birthday in the Feb. 10, 1981, issue of the Spokane (Wash.) Daily Chronicle: “Whatever, George, / Have a good one!”

The Saturday Review discusses its rising popularity (probably 1982):

As I said, Have a nice day seems to be dying out, at least around here. What we hear increasingly often is “Have a good one,“ spoken usually by young men. “Have a good one” is flip, cool, and cryptic. “Have a nice day,” wishy-washy or not, was specific by comparison.

Here's a couple of snippets of its use, both possibly from 1986. Here's more on "have a good day" and have a nice day" by The Grammarphobia Blog.

The Oxford English Dictionary shows "Have a good day" as far back as the 1200's ... then shortened to just "Good day" over time. (Similarly for "Good afternoon", "Good morning" and so on.)