I've made quite a few 'sound generators' on 8-bit (PIC and AVR) and there's a huge variation in complexity open to you, from the 6/8-pin PIC driving a piezo element to external flash and high speed PWM, over sampled with a 4-pole LPF to get fairly decent sound (10-12-bit, 16kHz sample rate, for recognizable music). To me this sounds like the lower end of a cheap small 8-bit micro, a transistor and a cell phone speaker.
I say first find a clip of the sound you want, preferably a clean one and then decide if you generate it directly or just play it from flash. SPI flash is pretty cheap these days, and as mentioned some PIC's have internal opamps for <$1 (Ian M mentioned the PIC16F1705, nice little jobs these are!) but the part count starts climbing. The idea of play sounds from flash has the benefit of being simple to implement,and allows you to play any sound of almost any length you want (limited by size of flash). The micro only has to periodically grab bytes from memory and update a PWM, which is great if it has to do other things, but seems a waste if that is its only job. I thnk unless you're playing high quality music, an external DAC is also over kill.
AVR's, specifically the attiny series almost always have PWM capability, and some have a high frequency oscillator, 32/64MHz to drive it, pushing the PWM frequency up and easing the requirement for high pole filters. Both PIC's and AVR's run fast enough to do fairly complicated maths for generating low sample rate 'sound effects' on the fly, but these would be one-trick ponies and wont' have the versatility of simply 'dumping a wave file in flash'.
So yeah.. can be as simple or as complicated as you want, and I suspect thats why you posted - if there was only a couple of ways to do it you would have done it
Would be nice to hear an example of the 'click' sound you want. Piezo's can 'click' nicely given the right waveform, but might not be loud enough.