Take a look at the BLE FindMe examples. On the button side, a button click starts scanning (because its a Central) and goes in and out of deep sleep. You can switch a couple of options around, but essentially, you can make it into a peripheral that advertises on button click, then goes into deep sleep after it times out, and only wakes up/advertises after a click.
are the two functions you would need. You will also need to handle the time out cyble event in the BLESS event application callback function.