A slight variation on the OP's original proposal/spec, but a fun project would
be to re-design/implement Babbage's analytical engine using relays. This is
something that could have been accomplished in 1900 - both the technology and
the concept existed - if only somebody had put 2+2 together at the time (pun intended).
It's interesting to speculate that if Babbage had been born or been working
just a couple of decades later than when he was, or alternatively if electrical
concepts had been just slightly more advanced in his time, he would have
throught of implementing his ideas with electric relays and would likely have
succeeded (although one might argue the same about the state of mechanical technology).