Hi digger
Have a look at the datasheet for the SN74278. 4 Bit Priority Regs, you would need two in cascade.
www.datasheetarchive.com
Use the outputs to drive a single 8 transistor i/c, ULN2003A series, as the relay drivers.
Couple of obvious questions. [ its not egg sucking time, I'm just interested]
You need an OFF button.
Do you want 'break before make', or 'make before break', switching ?
All relays 'OFF' at power up.
If you don't like the SN74278 route,I have an idea using low power SCR's as the relay drivers and a few other components.
Attached SCR1 dwg, sorry its a little rough, but it gets the method across. SCR's TO92 style, 100mA
Push any button, all relays RL1 to 6, OFF, release the button then the selected relay switches ON.
Break before Make.
Regards
EricG