Does the basic 2 x 4 brick have a common nickname?
It's called a brick.

This may sound like a weird or mocking answer, but when you think about it, it does make sense.

The 2x4 brick the most iconic brick of them all, the one which has started it all. The one for which the first patent has been issued in 1958. The book "50 Years of the LEGO brick" contains 6 of them, and the number of permutation for these 6 bricks has been the classic question to illustrate how endless LEGO construction can be (even LEGO got the answer wrong at first, by the way).

It's the one they use on the cover of collector's guides, it's the one they make bigger.

Face it, it's the brick.