Thanks, I think I just needed some encouragement.
I had some time to poke around and found some things:
Licking my finger and rubbing it between the resistor and electrolytic cap on the chip board changed the speed/pitch. It sounded good, but when I tried to recreate it with a pot, the unit wouldn't even produce sound.
Wiring a pot as a voltage divider with the middle lug to the same spot on the board gave me another type of speed/pitch change. I had a feeling I could do that one, I was just looking for other things first.
Adding a jumper from the left side of C14 (the cap in the pic with the longest legs) to the same spot on the chip board made for scrambled sound. Would probably be good with a momentary switch. If a drum beat is going and you make this connection, the beat will continue as soon as you disconnect it. I tried a pot and got some more pitch/speed change as well.
Adding a jumper from the jumper right in the middle of the last picture, to a lot of the pins from the chip create background drones. I think hooking it up to pins 10 and 11 (counting from the left) sounded good to me.
Adding another jumper from that same jumper I just mentioned to pins 1 or 2 (again from the left) made the drum beats and animal sounds louder and more clear.
Any other things I find, I'll update here.