My first reaction to that probelm would be to use an Arduino (or other 8 bit MCU) rather than effectively 16x 555 timers, even though the Arduino would need extra transistors to drive the relay coils, and a resistor ladder to put multiple buttons on each of the few remaining input pins.
However, you've madea start with the 8x 556 chips, so to give us a chance to figure out what's going wrong, we need to see the schematic, + good photos of the project - an overall view so we can see the general layout and what's hooked up where, and a closeup of each sde of the circuit board.
I suspect that your decoupling or layout is inadequate so when one timer pulls a current spike as it switches, its glitching all the others. However that's pure speculation until we've seen the schematic and photos.