Of American-English origin, the noun YIMBY (also Yimby, yimby):
– As an acronym from the initial letters of yes in my back yard: designates consent, by nearby residents, to the siting of something despite the fact that they perceive it as unpleasant or hazardous; this acronym also designates a person holding such an attitude.
– As an acronym from the initial letters of yes in many back yards: designates consent, by members of a community, to share the problems that may be created by the siting, in another community, of something unpleasant or hazardous that will benefit them; this acronym also designates a person holding such an attitude.

The noun YIMBY was coined after NIMBY, an acronym from the initial letters of not in my back yard, which designates opposition by nearby residents to the siting of something that they perceive as unpleasant or hazardous.

The earliest occurrences of the noun YIMBY (also Yimby, yimby) that I have found are as follows, in chronological order:

1-: From UCR chief cites growing toxic threat, by Ed Mauel, published in The Sun (San Bernardino, California, USA) of Sunday 23^rd February 1986 [page B1, column 1]:

Ted Hullar, chancellor of the University of California, Riverside, […] identified hazardous waste management—handling, disposal and reduction—as the toughest public policy issue facing America today.
[…]
The problem, he said, is everybody’s. Anybody who uses anything they didn’t grow probably supports a hazardous waste-producing industry somehow, he said.
The distillation of the problem, Ted said, is the Nimby Syndrome—“Not in MY Back Yard”—that needs to be replaced by the Yimbys Syndrome—“Yes, in Many Back Yards.”
Some community is going to have a waste disposal facility. The rest of us need to help that community deal with whatever deficits it creates there because it will be serving all of us, Ted said.

2-: From the following letter to the Editor, published in the Tri-City Herald (Pasco, Kennewick, Richland, Washington, USA) of Thursday 2^nd October 1986 [page A4, column 4]:

YIMBY, not NIMBY

Naysayers are getting to me.
This country was not built by people who said “no” to everything. It was done by people who said, “Yes, let’s try.”
With respect to storing radioactive materials, we’ve been doing that at Hanford for a long time, and we know how to do it better than anyone else. Let’s study the site to be sure it is safe. If it turns out to be a good place, let’s say YIMBY (yes, in my back yard) and put a stop to NIMBY (not in my back yard).
It’s time to be positive! YIMBY not NIMBY!
MILTON LEWIS Richland

3-: From the McHenry County’s Northwest Herald (Woodstock, Illinois, USA) of Thursday 22^nd October 1987 [Section B, Community: page 8, column 1]:

Recycling Round-Up
A schedule of recycling drives provided by the McHenry County Defenders
Yes, In My Backyard (YIMBY)

An innovative solution to the growing problem of solid waste disposal would be for each one of us to take responsibility for our own wastes, forcing ourselves to find long-term solutions which do not endanger our environment. National and state legislation should require that all wastes be disposed of within ten miles of where they are generated. This would apply to mines and factories as well as households and office buildings.

4-: From the following letter to the Editor, published in The Boston Globe (Boston, Massachusetts, USA) of Saturday 26^th December 1987 [Vol. 232, No. 179, page 22, column 5]—MWRA stands for Massachusetts Water Resources Authority:

MWRA director Levy’s invitation to the 351 Massachusetts cities and towns to nominate sites in their own communities for waste-water treatment facilities in return for adequate compensation deserves serious consideration.
The plan may not convert NIMBY communities into YIMBYs (“Yes, in my back yard”) overnight, but it’s a rare creative proposal from a conscientious public agency not afraid to tackle a problem that threatens to overwhelm us in our own lifetime.

5-: From How Do You Spell Trouble? For Business It’s N-I-M-B-Y, published in The Scranton Times (Scranton, Pennsylvania, USA) of Sunday 10^th July 1988 [page A21, column 4]—reprinted from Coping in the Age of ‘Nimby’, by William Glaberson, published in The New York Times (New York City, New York, USA) of Sunday 19^th June 1988:

Some companies are bringing community groups in early and trying to make their proposals seem palatable. At a meeting of local government officials in Orange County, Calif., recently, executives from Environmental Systems proposed building a toxic dump facility that could be moved on flatbed trucks. The company pledged that the entire operation would be taken to a new area in five years.
The company’s Morell has his own acronym answer to Nimby: Yimby, or “Yes, in many back yards.” And in one form or another many analysts are talking about similar ways of making people understand that they must share the risks that go along with the benefits of modern life. People, they say, will only take risks that seem to be justified by local problems or that are in proportion to financial or other benefits that can be gained.

6-: From Political Notebook, by Alan Janson, published in The Daily Breeze (Torrance, California, USA) of Sunday 11^th December 1988 [Vol. 94, No. 346, page A3, columns 1, 2 & 3]:

ELDER FOR CHIEF OF THE YIMBYS—Assemblyman Dave Elder of Long Beach is considering forming a new club of YIMBYs, a counterweight to NIMBYs.
NIMBYs are the common breed of politician (or common citizen for that matter) who decries a public problem, like the lack of prisons, but fights like hell to keep it out of his district (or neighborhood.)
They are people most accurately described by the phrase: Not In My Back Yard.
Now, Elder of Long Beach is proposing an Assembly resolution that denounces any congressional effort to close the Long Beach Naval Shipyard. A lot of jobs are connected to the shipyard.
“We become YIMBYs when it creates jobs in our districts,” he said.
Get it? Yes In My Back Yard.

Coined after NIMBY and YIMBY, the noun MIMBY is an acronym from the initial letters of maybe in my back yard, which designates a cautious but open attitude towards the siting of something perceived as unpleasant or hazardous; MIMBY also designates a person holding such an attitude.

The earliest occurrence of the noun MIMBY that I have found is from New generation struggles to emerge in Tucson elections, by Tim Steller, published in the Arizona Daily Star (Tucson, Arizona, USA) of Sunday 9^th April 2023 [page B1, column 5]—Val Timin, aged 32, was the new president of the Palo Verde neighborhood residents’ association:

A new generation of leaders, many in their 30s, is pushing to change how Tucson grows. […]
[…]
These new leaders may not be so pro-development that they belong to the Yes In My Back Yard, or YIMBY, movement. But neither are they traditional Tucson neighborhood NIMBYs—homeowners with a kneejerk negative response to proposed changes.
Timin described her approach as closer to MIMBY—Maybe in My Back Yard—open to proposed changes but still guarded.