Make sure the coil gap between the magnet on the fly wheel and the coil is right and that nothing is blocking it. Had one once that had some metal shavings stuck to the magnet. I cleaned the magnet off and every thing worked fine.
Check eBay for cheap coil knock-offs. I have a Honda GX120 motor on a power washer that had a faulty coil. Got the coil on eBay for $8 I think. Worked fine.
If your genny has a low oil cut-out, it may be faulty. (check oil first) You can bypass the switch by simply unplugging it. The low oil switch will kill spark if it is faulty.
The on/off switch may have failed. Check that too.