What's the origin of the phrase "cruising for bruising"?

From Newsweek magazine, 1951...

Boston has come in for its share of the new slang. In Back Bay or in Charlestown, on the hook is a boy or girl in love. Cruising for a bruising is looking for trouble ; a flookie is a jerk, noivice in soivice is jumpy, and loco in coco is just plain nuts.

I'm in the UK, and I don't recall hearing/reading this expression until around 2000, so my guess is it didn't really cross the Atlantic until social media / Internet chat facilitated more widespread dispersal.

I'd also add that in all the hundreds of times I've come across it since then, I've never known the article to be omitted as per OP's title.

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Note that (in BrE, at least) it's not used in quite the same way as looking for trouble. It wouldn't be unusual to hear either of these in a London pub today...

1: You're cruising for a bruising, mate! (threatening: Back off or I'll hit you!)
2: Don't go looking for trouble! (genuine advice)

...but you won't often hear either of those idiomatic usages in the alternate context.

The expression appears to be from Black English extension 1940s usage of cruising meaning strolling:

Cruising for a bruising: verb phrase

Looking for trouble; courting violence, esp while riding about in a car.

Origin:

1951+ Teenagers; perhaps from or influenced by black English cruising, ''strolling, parading,'' attested by1942.

(The Dictionary of American Slang)

Early usage example from Wiktionary:

1945 Nov. 30, "Underdog, Undaunted Navy Ready for Army," The Independent Record (Helena, Montana):

• Navy should score once but the Mid-shipmen definitely are cruising for a bruising.

According to Green’s Dictionary of Slang:

cruising for (a) bruising (also cruising for a smacking):

• [1940s+] (orig. US) looking deliberately to cause trouble.

the same Dictionary cites the following black English usage from about the same period:

Cruising:

7. [1940s] (US black) to walk someone along/around/through.

8. [1940s+] (US black) to walk in a strutting manner; thus cruising n.

The earliest instance of the expression that an Elephind search returns is from a very brief item in the “QM Quips” column of the [Camp Shanks, New York] Palisades (December 29, 1944):

Our best wishes to Sgt. Harry Lagerman and T/3 Johnny Williams on their new asgmts, also to Albert Hosier, Walter Smith and Roger Vadnais. . . . Don Ness is “cruisin’ for a bruisin’” these days. Too many girls.

As its location (Camp Shanks, New York) implies, the Palisades was a military newspaper.

Likewise, the story "Underdog, Undaunted Navy Ready for Army," in the [Helena, Montana] Independent Record (November 30, 1945)—quoted by Wiktionary and noted in user2922582’s answer—has a military angle, although a rather attenuated one:

Navy should score once but the Mid-shipmen definitely are cruising for a bruising.

The subject of this story is the annual collegiate football game between the U.S. Naval Academy and the U.S. Military Academy—a game that Army (the top-ranked college team in the country in 1945) was heavily favored to win. In the event, Navy did get bruised, losing the game 32–13.

J.E. Lighter, Random House Historical Dictionary of American Slang (1994) cites a 1947 supplement to Lester Berry & Martin Van den Bark, American Thesaurus of Slang as the earliest instance of the expression that Lighter is aware of:

cruisin’ for a bruisin’ {or bruise} inviting a beating; heading for trouble. [Earliest citation:] 1947 ATS (Supp.) 4: Cruisin’ for a bruisin’, riding for a fall.

The edition of The Dictionary of American Slang that user2922582’s answer cites appears to be the third, from 1995. But the earliest mention of the phrase in that series of dictionaries is from Robert Chapman, New Dictionary of American Slang (1986), which offers this slightly different entry:

cruising for a bruising v phr teenagers & college students Looking for trouble; courting violence, esp while riding about in a car

Earlier in the same series, Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first supplemented edition (1967) introduced a lengthy and fairly detailed entry for the verb cruise:

cruise v.i., v.t. 1 To walk, drive a car, ride a motorcycle, etc., slowly through a particular area in order to survey the general activity, social prospects, etc., esp. in search of a sex partner; to move about in a place, usually a crowded or public place, party, etc., in search of a sex partner; to look for a “pick-up.” 2 To make a subtle, tentative, usually unspoken sexual approach to someone.

The most detailed coverage of “cruising for a bruising” and its possible antecedents appears (again as noted in user2922582’s answer) in Jonathon Green, Slang Dictionary (2008). Green’s entry for “cruising for a bruising” has two parts:

cruising for a bruising {1940s+} (orig. US) 1 looking deliberately to cause trouble. 2 acting in such a manner that will get one into trouble, usu. of a physically harmful nature.

The entries here seem to me to be out of order, since the three earliest examples I’ve been able to find—from 1944, 1945, and 1947—unmistakably use “cruising for a bruising” in the second sense that Green records. Even the 1951 Newsweek example that FumbleFingers’s answer cites (“Cruising for a bruising is looking for trouble”) unquestionably an example of Green’s definition 1 in action, since it can be read either literally (the person is trying to find trouble) or idiomatically (the person is headed for trouble, intentionally or not).

But Green’s entry for cruise as a verb is especially interesting. Here are some relevant portions of it:

cruise v. {fig. uses of S[tandard] E[nglish] cruise, to sail to and fro with no particular destination; note Ned Ward, Hudibras Redivivus (1705–7): ‘Now gently cruzing up and down, / T’observe the Follies of the Town’} 1 {late 17C+} to approach someone obviously with sexual intent, both for commercial or non-commercial purposes ... 2 {mid-18C} (UK Und.) to beg. 3 {19C+} to wander along/through. 4 {20C+} (also cruise around) to drive around, often along a town’s main street, surveying the situation, looking for friends, men/women to pick up etc. 5 {1910s+} to search for sexual contacts by walking specific streets, areas etc. 6 {1940s} (US black) to walk someone along, through. 7 {1940s+} (US black) to walk in a strutting manner. ...

Perhaps most striking is the similarity of Green’s definition 4 of cruise (dating back to “20C+”) and Wentworth & Flexner’s 1967 definition of the same word. But more broadly, we have evidence of the notion of “cruzing up and down” in order to check out a scene going back to 1707. Under the circumstances, it hardly seems necessary to suppose that teenagers in 1951 started saying “cruising for a bruising” because “cruising” was in use in black English by 1942 in the sense of “strolling, parading.”

My own speculation is that “cruising for a bruising” arose at some point in the early 1940s as an offshoot of cruising in the “checking out the scene and discreetly searching for a sex partner” sense—and then very quickly acquired a more general sense of “looking/asking/heading for trouble.” Don Ness of Camp Shanks, New York, in December 1944 may have had an innocent interest in girls—but associating with too many of them was, in the newspaper columnist’s eyes, prima facie evidence that he was “cruisin’ for a bruisin’.”

I have nothing to offer other than my experience from the phrase in the 1980s living in Louisville, KY as a child. My mother would say ... "you're crusin for a bruisin" and I knew that I was close to getting hit if my behavior didn't change. Lovely that she gave a warning.

But I'd agree with the notion that it does not mean "look out" or some careful warning ... it is a threat that you need to back off.

So Boston? I have no idea...my mother never lived in Boston ... her parents were from Appalachia and Panama, Panama. She was born in the 1940s though so maybe it has some baby boomer connection.