I would consider using these palettes:
4 colors:
Black Red Cyan YellowBlack Red Green Yellow16 colors: ZX Spectrum palette with values 0, 170, 255. In other words, RGB combinations using 0, 255 or 0, 170.
256 colors:

This one is a subset of 9-bit RGB palette of 8 values for each, with LSBs of components merged like this:
8bit 9bit (rgb)
00 000
01 011
10 100
11 110
with the only exception being brightest color, which is white. Available pure colors are black, red, cyan, yellow and white. There are 5 true grays.

Subset of 12-bit RGB palette where 2 LSBs of components are merged. 16 grays available.
65536 colors:
A palette with 4 bits of red, green, blue and alpha (16 grays).
Another one:

4 LSBs of components are merged from 24-bit RGB. 256 grays.