It wasn't a design project but it was the project that first got me interested in electronics. Up until then, it was astronomy that had me.
When I was about 8 (around 1977), my dad bought a kit for the Sinclair Scientific calculator and he taught me to solder as we built this kit together.
The Sinclair Scientific was pretty revolutionary at the time: it was small, having been designed around a reprogrammed Texas Instrument TMS0805 chip, and had a bright red LED display that displayed numbers soley in scientific notation - 5 digit mantissa, 2 digit exponent. It had 4 scientific functions and used reverse-Polish notation (RPN).
I thought it was the coolest thing ever at the time. It was still working in the early 2000's but I lost track of it at my parent's house, so it may still work...
A few years after the calculator my parents got me a kit for the ZX81 and that was me hooked on computers.