I've been thinking about this, too, as quite a few young students are finding me recently-- word of mouth from a few happy parents. Most of them are doing well, but I feel like I'm in uncharted territory.
I think I have 5 students under age 7. With two of them, I haven't been using a book other than a manuscript notebook. Lately I've been helping them compose little melodies, usually 8 bars long, as I write it down. I might start by saying "lets write a song using three notes: C,D,and E". The following week I might do another song with C,D,E,F. Next week, CDEFG. I like to do it starting with A, too, feeling that it's easier to find interesting sounds in a minor key.
Some of them take right off playing mayhem with those three or four notes. Others need encouragement: "play any old thing" or "why don't you start with C?". I usually have to joke around about not being able to write it down if they play anything too complicated, and would they please play slower and keep it simple? I usually get one phrase at a time (maybe 3 to 8 notes), write it down, and then ask them to either play it again, or if the momentum is there, to play the next phrase. I might have to take some liberties writing the rhythm to make it all quarter-, half-, and whole notes (unless they play something REALLY cool). If they play a phrase that offers a cadence, I might just grab that, and leave out some notes that they played after. I often have them go back and play what we've already written down, just to keep the sound in their ears as we progress.
I sometimes make suggestions as to what direction they might take, like that they might want to repeat a phrase, or that it might sound good to end on C.
When we're finished, I ask them what it's called. This is the hardest part... "Dinner at Katie's", "Swimming", or "Lilly's Green Sweater". Then I have them write "by studentname" after the title.
Sometimes, despite my subtle or not-so-subtle direction, we fail to come up with something with cadences that make sense... maybe it stays on the c major chord the whole time or something. They can still practice it and be proud of it, but I think the coherent musical ideas seem to get practiced more. If a tune like that isn't getting practiced, I don't dwell on it. Often they learn it by rote as we write it.
I might harmonize the tune and play chords (I IV and V chords) along with the student. I will probably teach them the chords (all white notes "just make this shape").
Just in case I sound authoritative, let me repeat that I'm also figuring it out as I go.

I'm really interested to hear what other people are doing.
m1469: thanks for the insight about the general plan... I've never come up with an explicit structure, but it sounds like a really good idea.