The parts you need to add an AC adapter are fairly simple, but harder to find than I expected. I ended up going to Radio Shack -- they carry specific-voltage adapters, individual tips, and panel-mount sockets to match the tips. At most places you have to by a "universal" adapter, and can't get the sockets at all.
For the TinyLove mobile, which uses 3 AA batteries, I bought
4.5V adapter (700 mA)||
Size "M" (2.1mm ID, 5.5mm OD) plug||
Size "M" panel-mount socket with switch||
For your application, choose an adapter with an output voltage that matches the type and number of battery your device uses.
You will also need lengths of thin (24 gauge or smaller) wire, one with black insulation and one with red. I cannibalized mine from some left-over four-conductor signal cable. Start with 6" lengths and trim them back when doing the soldering (Step 5). A wire stripper will be quite handy, but someone with some skill (not me) could use some small wire-cutters or scissors.
You'll need some tools: The TinyLove mobile is held together with weird triangular-bit screws. I found the bits at McMaster-Carr Industrial Supply
(items 5941A11 through 5941A14).
A 21/64" bit will make the exact hole needed for the panel-mount socket. You could also use a 1/8" Dremel bit, followed by the Dremel ball grinder to enlarge the hole.
A soldering iron, solder and flux are needed to connect the wires. A solder-sucker might be helpful, too, when disconnecting the leads from the battery terminals.