The informal phrase (as) thick as a brick means very stupid.

In this phrase, both the rhyme and the humorous comparison with a brick give emphasis to the adjective thick, meaning stupid.
—Cf. the synonymous phrase (as) thick as two short planks.

The phrase (as) thick as a brick was popularised by Thick as a Brick (1972), the title of an album by the British rock band Jethro Tull. The lyrics by the British musician and singer-songwriter Ian Anderson (born 1947) contain the following two lines:

Your wise men don’t know how it feels
To be thick as a brick.

However, the phrase (as) thick as a brick predates Jethro Tull’s album. The following, for example, is from an interview of the British actress Judy Parfitt (born 1935), by Terry Coleman, published in The Guardian (Manchester, Lancashire, England) of Saturday 23^rd August 1969:

Tony Richardson was doing “Hamlet” at the Roundhouse, he asked her [to play Gertrude], and she said it was the most stupid idea she ever heard of in her life, did she look like Nicol Williamson’s mother? But then Tony—and here she does a very creditable take off of Mr Richardson—said he saw Gertrude as sensual, sexy, snake-like, a woman.
Oh she said, and went away and thought, and then did it, though she had never done Shakespeare before, never not once, and Gertrude turned out to be a dreadful part. Act I, says Miss Parfitt, she does nothing. Act II—“Look where he comes,” and that’s it. Then the close scene, the big one. Then Ophelia drowns: “There is a willow, blonk blonk.” And then you die.
But as she played it, I asked, because on this depends the whole playing of the part, does Gertrude know Claudius murdered the old king her first husband? “No, she doesn’t. She knows Claudius is good in bed, and she never thinks, may be until the play within the play, and there are the two kings and one queen, and she begins to think to herself, s’funny she’s as thick as a brick.”

In early use, the phrase (as) thick as a brick was applied to nouns such as skull and head, used metonymically for intelligence. The earliest occurrences of this that I have found are as follows, in chronological order:

1-: From The Democratic Star (Helena, Arkansas, USA) of Thursday 27^th September 1855:

Carrying out the Principle.—By a late number of that consistent and able democratic journal, the Chattanooga Advertiser, we see that twenty-three bigoted and silly know nothings have withdrawn their patronage from the concern […]. They charge the editors with advocating abolition doctrines in an article on the White Basis. We have carefully read the article in question, and cannot find a single abolition sentiment in it; nor can we see the good policy of the very silly course of these very silly know nothings. If they cannot see that such a course as this is better calculated than any other to build up the paper, they must have skulls as thick as a brick!

2-: From the Hammond Gazette (Point Lookout, Maryland, USA) of Wednesday 30^th December 1863:

New Year’s Oration.

My delectable, supereffiminate and conscribed secesh auditory, lend me your ears for the short space of no time and I will endeavor to substantiate, equivocate, prevaricate and expluvicate my idears right down on your capputs as thick as bricks in a meetin house, in prose, verse and rimes, bout bad peple and kurius times.

3-: From Purity Par-ity! A New Version of “Humpty-Dumpty”, published in The Fife Free Press (Kirkcaldy, Fife, Scotland) of Saturday 19^th November 1892:

Sleek Purity Par-ity, now grown thin and tall,
Thought he’d try to be master up in the Town Hall,
And the old process of treating was tried, as a trick,
Tho’ the heads that were treated were “as thick as a brick.”

4-: From the Kansas Topics (Kansas City, Kansas, USA) of Friday 25^th October 1895:

We claim that two of the Kansas colored men who run their own papers cannot edite [sic] themselves and they are the American Citizen and the Topeka Call. Their skulls are as thick as bricks.

5-: From an interview of ex-inspector Flynn on his thirty years’ experience in the Swansea police force—interview published in the South Wales Daily Post (Swansea, Glamorgan, Wales) of Wednesday 27^th November 1895:

“In my opinion it is not necessary for a superior officer to drill the men, because there is a drill-instructor for the purpose, but when men are required to be moved about for any purpose in large or small bodies in the case of processions or strikes then it is necessary that they be properly handled by a superior officer. Under such circumstances you could not expect a drill-instructor to move the men about, and I say emphatically that a man who cannot handle a body of men after being himself in the force nearly a life time, must have a head as thick as a brick!”