Not really "off the beaten path," as it's just a short walk from the Östermalm subway stop and Stureplan, and it's a convenient stop before or after the more energy-draining trip to Djurgården, but the Stockholm Music Museum has been an old standby of ours since Bella was a baby. Two whole sections are dedicated to trying all sorts of acoustic and electronic instruments, and throughout the museum, there are headphones to put on and buttons to push, perfect for compulsive little button-pushers and dial-fiddlers like Bella. And, unlike many "high-touch" museums, everything is well-maintained and 99% of the instruments and equipment actually work.
And this is fun; Bella at the same instrument in 2006:
Musikmuseet, see their website for current opening hours.
Sibyllegatan 2, behind the Royal Dramatic Theater ("Dramaten")