[PlunderBunny]

I believe the reason that 111 was chosen in New Zealand was because, on the analogue network, '1' was a 'long pulse', where-as '9' was a 'short pulse' (think about how long the dial takes to rotate back to its start position for each number). It was considered more likely that telephone cables swaying in the wind and touching would trigger emergency calls if the emergency number was 999.

Edit: Corrected to 111 for New Zealand, as helpfully pointed out.

[calfuris]

Brits went with 999 because it was easy to convert payphones from allowing 0s to be dialed for free (allowing users to call the operator) to allowing 0s and 9s to be dialed for free. In the UK, the system was that X pulses would be sent to send the digit X (10 for 0). NZ was using British exchange equipment that would recognize three sets of nine pulses as the emergency number, but NZ dial phones used the reverse convention (10-X pulses) so that mapped to 111.

[guidedlight]

000 is used in Australia 111 is used in New Zealand