We have sugar now, so yesterday I pulled the starter out of the fridge, fed it a spoonful of sugar, and let it sit to see if it would work out or if we needed to start over.
This morning, it's bubbly again! It is risen!
We're starting with 1L, rather than a full gallon, to see how it works. Since all the recipes I've found are for a gallon or more, we had to do some math.
There are approximately 4L in a gallon (rounding up. It's actually 3.785. But for an inexact experiment like this, we'll treat the liter bottles as quarts and be done with it).
1 gallon needs 1.5 cups of sweetener. I didn't ask LemurBoy to do that one in his head (0.375 cups, or a bit more than 1/3 cup). We dissolved that (minus a tablespoon or so that LG spilled on the counter trying to dump it in the wrong place...) in 2 cups of water, added the flavoring (one bottle contains 2 fl oz and makes 4 gallons, so 1/2 fl oz makes 1 gallon, so 1/8 fl oz makes a quart. Google tells me 1/8 fl oz = .75 teaspoons), and brought it to a boil, partially to aid in dissolving the sugar, partially to kill off any microorganisms already in the water. I asked LB why we were boiling the water, and he guessed the microorganism part.
While letting that mixture (called "the wort") cool to body temperature (LB correctly figured out that this was so it wouldn't kill the good bacteria in the starter), we put the starter in the bottle (1 quart per gallon, so 1 cup per quart), and stuck the rest back in the fridge, to be resurrected again if this works out well.
After adding the wort to the starter, and adding a bit more water to fill it out as there were only 2 cups in the wort (we probably should have boiled this), we put a balloon with holes poked in it over the top. LB again figured out why with no problem - the balloon keeps environmental microorganisms from getting in, and the holes allow some of the gas produced by the fermentation process to escape, so the balloon (or bottle!) won't explode. (While there is some alcohol produced in the fermentation process, it's a very small amount if you don't let it sit too long, and should be fine for kids in reasonable quantities. Just be sure to taste it yourself first to be careful.)
Again: We're using well water. Regular city tap water needs to either be filtered or left to sit for a few days to dechlorinate or the good bacteria won't grow.
And now we wait! It's supposed to sit on the counter for a few days, tasting it every day to see if it's ready, and cap it and put it in the fridge once it reaches an appropriate balance of sweetness and fizziness (the bacteria eat the sugar, so the final product should be significantly less sweet than it is now). It's supposed to be a little on the sweet side when bottled, because it needs to ferment slightly further after that to pressurize before being refrigerated.