Since you will really only have one switch input at a time, I would use an 8-bit priority encoder (or more than one if you have more than 8 switches) such as the 74HC148. This will take the 8 inputs and reduce them to just three bits with the bonus of having a signal to let the arduino know that a switch was pressed. These can be cascaded to provide more than 8 inputs.
And for the LEDs, just do the opposite: use a 74HC138 which is a 3 to 8 demultiplexor. 3 signals will control 8 LEDs.
Together these reduce your pins from 16 to 8 (for 8 switches and 8 LEDs)