Toggling series of memory cells in minecraft with one button press

Solution 1:

Probably the easiest and most robust approach (using actual memory cells is prone to entering illegal states like 2 cells active) is to use the N-state T flip-flop design; in this case 12-state. It's a simple loop of interleaved droppers and hoppers, activated all at once by a common line of redstone dust. There's one item in the loop, and a comparators indicates in which of the droppers it is. Upon pulsing the dust with redstone signal, the item is shifted to the next dropper (ejected into a hopper on rising edge, inserted into the dropper after falling edge).

As long as you maintain the topology, you can take liberties with the geometry - make it into 2 parallel lines of 6 pairs, or snaking back and forth; you can also insert more hoppers (at cost of small delay to the shift). Skipping hoppers or adding droppers without hoppers is asking for trouble though; the update order may shift the item an arbitrary number of droppers on one activation without the hopper buffer present.