Used in this sentence (by a friend):
Well, if it doesn't work, just bucket and chuck it.
The earliest Google Books match for "bucket and chuck it" appears to be this one from Michael Bouquet, Westcountry Sail: Merchant Shipping, 1840–1960 (1971) [snippet not visible in window]:
The helmsman stands on a grating with his back against two teak rubbing boards on the wheel shelter. To port, the wheel shelter houses the lavatory (of the bucket and chuck it variety); to starboard, a lamp store and paint locker.
This is a boat-related example, but there is also this example from Suzanne Beedell, Water in the Garden (1973) [combined snippets]:
...just dip out the water with a bucket and chuck it around the garden, for even a small pool holds an awful lot of water (one of 6ft diameter and 15in deep holds 250–270 gallons).
If you are lucky enough to have a clean stream running through your garden, then a small paddling pool with a continual flow of water is not too difficult to make. There is no problem about filling and emptying and keeping the water clean.
Another early match is from Paul McClory, How to Use Natural Energy (1978):
When Paul and Mary Garton bought their Suffolk cottage the only sign of main services was a brass cold water tap. The wattle and daub structure, with Tudor brick floors, had been "modernised" around 1800. "The old folk here drank the water from the pond until a few years ago", said Mrs. Garton as the builders worked on the renovation, "and sanitation was of the bucket-and-chuck-it-variety".
Also, from Maurice Cloughly, A World to the West: A Voyage Around the World (1979) [combined snippets]:
For several days we made poor progress in changeable weather while a variety of things on board went wrong. The toilet started playing up so we were forced for a time to resort to the time-honoured procedure of "bucket-and-chuck-it." Seams in the mizzen lost their stitching. Jib sheet blocks tore loose out of the track.
Another early match appears in Larry Brown, Sailing on a Micro-Budget (1985) [combined snippets]:
Finally we come to everyone's favorite, the marine toilet. Coast Guard regulations forbid toilets that discharge wastes over the side. I find it strange that yachtsmen, who claim to love the sea, take such perverse pride in flouting the intent of this law. The law is worded in such a way that it is legal to go sailing with no toilet. Many sailors carry a sweet-smelling cedar bucket and operate on a "bucket and chuck it" philosophy...
And here is another from a non-nautical setting, in Association of Applied Biologists, Field Trials Methods and Data Handling: 16–18 December, 1985, Churchill College, Cambridge (1985):
The variability of the performance of pepperpot techniques for granules and hand application ('bucket and chuck it') for fertiliser, especially between seasoned hand and student has been noted and criticised by the farmers. No doubt a set amount of product has been dispensed over an area, but where, and how long has it taken? What is the coefficient variation with these techniques?
The "time-honoured procedure of 'bucket and chuck it'" may well have originated (as Josh61's answer suggests) in the context of boats with limited waste-disposal facilities on board—but it's not impossible that, in the days before indoor plumbing became commonplace on dry ground, "bucket and chuck it" might have been used to describe a waste disposal method used in lieu of plumbing or inconvenient outdoor privies.
It is also interesting that the same phrase is used for water distribution in a garden (1973) and for fertilizer distribution on crops (in 1985). Evidently, the human urge to gravitate toward rhymes involving bucket doesn't end with "There once was a girl from Nantucket..."
- Supposedly from 'beakhead' part of the forecastle used by sailors as a toilet.
(A B Sea: A Loose-footed Lexicon Di Jack Lagan)
The routine was to 'bucket and chuck it'. You crawled to the bow end of the boat with baby wipes and the blue bucket. It was important not to confuse this with the yellow bucket – which was for washing up. Then you had to lower your shorts, ...(Wild Waters in the Roar Di Mike Noel-Smith)
A The sanitary system had a simple name: "Bucket and Chuck It." The Jeanie Johnston's waste system is highly sophisticated, employing the latest technology presently used on ships and airplanes. (Jeanie Johnston: A Voyage Against All Odds)
Simple containers are modern variations of the traditional 'bucket-and-chuck it' toilets, aiming to do little more than collect our deposits for composting. (Building Services Journal, Volume 20,Edizioni 7-12)