You can use a large number of repeaters to delay a signal but your best bet is probably building a monostable circuit: LINK
With one of these you can prolong the signal from a button for a good length of time and have it reset after, say, 5 seconds.
Alternatively you could use RS latches (exactly like monostable circuits but without the self-cancelling repeater string, see the Reddstone Guide I linked to above). Have a button press trigger and RS latch that opens the path you want to take and then put a detector rail on that path that resets the RS latch after you've taken it.
Thanks! I ended up doing a variation on that, but no reset with a detector rail at this point. I'm not sure that's feasible/necessary. Instead each button resets the system and turns that one on.