This thread has changed from a take-it-apart project to lots of advice on how to fix a Sonicare and put it back together. I hope all the great comments below help people bring their toothbrushes (or lockpicks or glass etchers or whatever) back to life.
Step 1: Overview
Circuitboard with on/off switch, two AA size rechargeable batteries under.
Step 2: Top
This is the top coil, which controls vibrations to the toothbrush head. I know the head has magnets on it, but how does it work...?
Step 3: Getting Ready to Remove Solder
Here are the 8 points that have to be removed from the base. The 4 in a row are the connections for the top head; the 2 under that are the battery power leads. The bottom 2 are the recharger leads. Note all the corrosion...
Step 4: Finally!
Getting all the points loose was difficult, partly because many of the leads had wires soldered to them below the circuitboard and it seemed to replenish the solder I removed. I heat sinked with alligator clips and paper clips in the tighter spots.
Step 5: Circuitboard and Battery Pack
lines show corresponding leads.
Step 6: Corrosion
I tried to get it off with a toothbrush (don't tell my girlfriend!) but had to scrape it instead. Here's what it looked like...
Step 7: Semi-cleaned Circuitboard
And here's what the board looked like after I scraped off the corrosion. It looks like I might have damaged by overheating while trying to remove the solder, too. Beyond hope? If I cleaned it would it still work?
Step 8: Just for Anyone Who's Curious...
Here's what's under the blue capacitor (?)
Another note for anyone who's interested in changing the batteries: they are a standard size but they are epoxied into the base with the same stuff that holds the coils. Remove them by cutting away the plastic film surrounding the cells, remove the cells, and then pull out the plastic.