"Free" Stopover
 

What exactly is a free stopover. My fare rules say I can have a free stopover in SFO on the way home, but if I do a flight change to stop in SFO, it certainly isn't free

I've booked this
Segment 1 CHC-LAX
Segment 2 PSP-SFO-AKL-CHC

This was a very different price (at booking time), for the same flights on the same day.
Segment 1 CHC-LAX
Segment 2 PSP-SFO
Segment 3 SFO-AKL-CHC

Also if I attempt to make the change now, its also a very different price (again, even for the same flights on the same days)

My fare rules say
GENERAL RULES TEXT
4 STOPOVERS PERMITTED ON THE PRICING UNIT
LIMITED TO 2 FREE AND 2 AT NZD 100.00 EACH.
2 FREE IN SYD/MEL/AKL/LAX/SFO/HOU
2 AT NZD 100.00 EACH.
CHILD/INFANT DISCOUNTS APPLY.

There may not be availability for whichever flights you're trying to move to create the stopover on the date you're trying to move the flights to. Have you checked other days? E.g., try three day stopover instead of two?

This is exactly correct. Free stopover means just what it sounds like, but you have to have availability in the correct inventory buckets or the fare can't be used.

OP stated :

Which most likely means that there's a married segment aspect involved.

I read that, but I didn't understand it, as you can't add a stopover if the flights are on the same day.

If OP is, as you suggest, simply using multi-city search, then, yes, you can get different prices based upon how you search, but that doesn't have anything to do with stopovers, because there wouldn't be a stopover.

If you’ve checked availability via ExpertFlyer / etc, try calling and asking them to price the change. The website often does a poor job of pricing complex itinerary changes.

Changing flights will reprice some or all of the itinerary. You can't just change to add a stopover later without consequence.

A free stopover is exactly that - you can add a stopover at any of those cities while using that fare component, and there is no additional charge.