Hi,
Usually (almost always) I'd recommend a wonderfully complex DIY approach. But getting audio quality up to spec, better than crappy 8bit 8khz (phone quality?) I'd say go for a cheap MP3 player. They're so cheap now days, sure you won't have your rotary switch, but they go for about 15UKP for a small one these days, and most have recording capability.
Failing that, there are ASIC's that will do that job, some have built in memory, others will require external. The sound quality is pretty weak IMO, but it'll be a whole lot easier than programming a micro with ram, flash, and a ADC/DAC.
Obviously if you use a small speaker, then sound quality won't really need to be good, because a small speaker would have a limited frequency response.
So, the mp3 player idea aside...a nice kit that will handle the audio and storage, and maybe add your own simple switching mechanism, be it with a rotary switch, simple logic, or a microcontorller.
some links:
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The Jaycar one looks great, the chip it uses allows multiple messages to be accessed seperately - may require modification of the kit, but should work quite well. Plus its cheap $40 AUS... dunno what the exchange rate is though
There are some ISD chips like the ISD2540 which would do what you want. But the above kits would work out (especially the 2nd). All you would need is a rotary switch, as you suggested and a button.
Good luck,
Blueteeth