Audio delay
As elektrodians suggested, there are many ways to do your project.
Key words: bucket brigade, analog delay line, ...
If you need high quality go 16 bit A/D , FIFO, D/A. Complex.
For medium quality use the much simpler analog delay line.
Those are used in guitar pedal effects. Around US$120.
Look into Radio Shack (and Digikey, etc) website for analog delay line chips.
FIFO s used to be very easy to build with SRAM and FIFO controller chips, now discontinued.
I can explain a little more on how to build a FIFO using SRAMS, if you're interested.
j7