I built one a long time ago. It was a lot of fun to play with. One day I
absent-mindedly held an old telephone exchange relay coil too close to
it, and inadvertently built a step-up transformer and burnt my thumb
with the arc - I didn't do that again.
I don't think very high inductance is necessarily what you want. If you
want to fire rings well, you want more voltage per turn, which means not
too many turns, but if there are too few turns, then the inductance may
be too low, (especially if the core saturates but maybe even if it
doesn't), and you may trip a breaker. From memory I think the coil I
made was with 1.7mm diameter wire, wound onto an old spool about 100mm
long, 100mm diameter. The core was about 35mm diameter and 400mm long,
made of a bundle of welding rods stuffed tightly into a PVC pipe.
For the coil, make sure you use enamelled winding wire, not ordinary
insulated wire, as the insulation needs to be thin in order to allow the
copper to be thicker (for a given overall wire diameter) and the
resistance to be as low as possible for a given number of turns. This
will minimise heating, though it will still only stay cool enough for a
few seconds at high power. If you want to run it for longer (which is
definitely interesting), use a variac or wire it in series with a heater
to reduce the current to a value that won't cause overheating.
You might find that for the core, dull steel wire for oxy welding is
better than shiny copper plated steel wire intended for TIG welding,
because the dirt and oxide on the surface of the oxy wire will help to
reduce eddy currents. The thick plastic coating of florists' wire may
use up too much space and reduce the iron that you would fit into the
core. The voltage between iron wires isn't much so you don't need very
good insulation, just a bit of dirt will do. If you were really
interested in finding the best possible material then there are special
materials intended for magnetic cores, such as grain oriented silicon
steel, but I don't think using these would add much to the educational
experience, and you could spend the effort instead on constructing other
models afterwards such as a linear induction motor:
https://www.youtube.com/watch?v=vxrUR_zZcmU
I recall that some of Eric Laithwaite's books contained instructions for
building the electrical machines. I would not pay too much attention to
his writings about gyroscopes though.