I've decided to add a buck converter to the board to make the 5 volt power supply from anything from 7-15 volts input. This makes the clock a lot easier to power. It's a little like building a "pi power" directly in, though the circuit will be slightly different (it'll be a copy of the 5 volt supply from my FE-5680A GPS board) because it's not going to support as much power. The supply will be rated for around 500 mA, which should be more than enough for the Pi Zero and clock display running flat out. Pretty much any power source that supplies 5W of 7-15 volt power on a 2.1mm center positive barrel connector will do.
EDIT: I've changed my mind. I'm going to just go with a 5V input jack like the GPS clock and my GPSDOs. I've got a bunch of those and adding a 5V buck converter to the board just raises the cost/price.
It wouldn't take much to *also* make a Pi Zero clock into a GPS stratum 1 server at the same time. To get it to work, you'd need to disable the serial console once you got it working on the network. Next, you'd connect a GPS receiver breakout board up to pins 4, 6, 8, 10 and 12 of the GPIO header (+5, gnd, TX, RX and PPS respectively). The easiest way to do this would be to use a "double male" stacking header (like this one) for your GPIO so that you could connect up to the top as well.