The phrase a sandwich short of a picnic and variants mean mentally deficient, slightly crazy.
—Cf. also white ants and not to have both oars in the water.

The earliest instance of a sandwich short of a picnic that I have found is from Penn & Teller: no label on their bag of tricks, about the American magicians and entertainers Penn Fraser Jillette (born 1955) and Raymond Joseph Teller (born 1948), published in the Philadelphia Daily News (Philadelphia, Pennsylvania) on 15^th May 1985:

As Penn explains to the packed WestSide Arts Theater, where the two of them have been billeted since mid-April, they are “a couple of eccentric guys who know how to do a couple of cool things.”
Cool things like juggling sharp knives, swallowing both fire and No. 6 embroidery needles and levitating cooperative members of the audience. Cool things like mewling, grunting and aiming a waterfall of pre-chewed apple at the front row. “Note the sound effects proving that the little self-respect I once had has by now completely fallen away,” says the imperturbable Penn in the midst of a juggling riff. “This isn’t a very good piece, but I bother to learn it and you’re certainly going to watch it.”
You watch, you watch wondering all the while if these guys aren’t maybe just a few sandwiches short of a picnic.

Generally, with words denoting some specified deficiency in a desirable or standard quantity of something, short of a —— means mentally deficient, slightly crazy as in a brick short of a load, a few slices short of a loaf, sixpence short of a shilling, etc.

This is only one of the manners of expressing this idea; the following was published in Column 8 of The Sydney Morning Herald (Sydney, New South Wales, Australia) on 14^th January 1986:

A Merewether reader is intrigued at the number of expressions Australians use to describe those that are “a bit slow, a bit thick — not the full quid”. “It would be churlish of me to suggest that with the changes taking place in education these days these expressions may come more into use,” our reader said. Our reader volunteered: “His elevator does not go all the way to the top”; “Two bricks short of a load”; “The lights were on but no one was at home” and “Not the full bottle”. Any more?

The next day, the same column had:

Column 8 was inundated with Australian expressions for those who are “a bit thick” (Column 8, yesterday). Food analogies were popular: a sausage short of a barbie ¹; a sandwich short of a picnic; the butter’s slipped off the noodles; not the full tin of Milo ² (or jar of Vegemite); a few slices short of a loaf; out to lunch; not the full tin of bikkies (or box of chocolates); and wouldn’t know what day it was unless he had two baked potatoes on his plate. Money also featured: only two bob in the pound; 60¢ in the dollar; and he’s the full quid, but it’s all small change.
Others included: a few roos ³ loose in the top paddock; not too tightly-wrapped; a few spanners short of a toolbox; a shingle ⁴ short; not all there; not playing with the full deck; a couple of buttons missing; not running on all cylinders; a couple of ribs short of a cage; a few bricks short of a load; his river doesn’t run all the way to the sea; hasn’t got them all in a row; hasn’t got both oars in the water; a bit of light gets in; not the full tube of Dencorub ⁵; if brains were dynamite he couldn’t blow his hat off — and many, many more . . .

¹ barbie: a barbecue
² Milo: a chocolate and malt powder that is mixed with hot or cold water or milk
³ roo: a kangaroo
⁴ shingle: a rectangular wooden tile used on walls or roofs
⁵ Dencorub: a brand of pain-relieving creams and gels

On 1^st December 1987, The Guardian (London) published a letter by a certain Kenneth Corn, “a bona fide American”, explaining an American-English phrase:

“Rowing with only one oar in the water”, simply means that the person has a mind that is going around and around in circles yet getting nowhere.

A reaction was published on 3^rd December:

Sir,—Surely Kenneth Corn’s explanation (Letters, December 1) of the phrase “rowing with only one oar in the water” is too elaborate: I took the phrase to mean simply a deficiency in the marbles department, e.g.: “He’s not playing with a full deck”; “Kangaroos in the top paddock”; and “Not quite enough coupons for the coffee percolator and matching set of cups”. — Yours,
Muir MacKean.
London W2.

And on 7^th December, The Guardian published this interesting letter:

Out to lunch

Sir, — Muir MacKean’s “kangaroos in the top paddock” (Letters, December 3) is a new one on this office which prides itself on collecting daft sayings for those “a few sandwiches short of a picnic”.
Current favourites are, “not got all their chairs at home”; “their elevator doesn’t go all the way to the top floor”; “not the full shilling”; “one short of a six-pack”; and “a few points short of promotion”.
The beauty of these expressions is their adaptability and topicality; hence, “a few points short of a council flat” or “one condom short of an orgy”. I could go on, but ’tis a queer bird the fish, as they say in Wigan. — Yours,
Mike Hill.
Manchester.

The following is from The News Journal (Wilmington, Delaware) of 24^th November 1988:

Moose, a barber at Boney’s Barber Shop in South Wilmington, says he eagerly awaits the annual college basketball predictions, especially those reserved in this space.
His enthusiasm lies in selecting against the teams favored by this reporter.
The mind of a college basketball forecaster can be described in several ways:
The cheese fell off his cracker a long time ago.
He’s a couple sandwiches short of a picnic.
He has one oar in the water.
His flag is at half staff.
He got his picks by dialing the toll free number: 1-800-NUM SKUL.