The ST90 has battery-powered ignition, and the bike needs a charged battery to make a decent spark at kickover. A good 6 Volt wet cell battery will measure 6.3 Volts fully charged, and a few tenths of a Volt higher if the "topcharge" has not been used up.
First test, check for DC Volts at the points. Using the kickstarter, rotate the engine through until the points open (the points should be gapped to .012") and measure the DC Volts. This test is done with the ignition switch ON and the kill switch set to "RUN". Measure DC Volts between the engine block and the part of the points that moves. This should measure slightly less than the Voltage at the battery.
The points have to be super clean. I use electrical contact cleaner made by Caig, however brake parts cleaner is pretty good. With the points closed, drag a piece of clean paper between the points to burnish them clean. Check the gap by opening the points and using a clean feeler gauge between them.
Ground connections give trouble on these bikes over time. The battery negative wire terminal is "grounded" to the chassis at a bolt. The ignition coil and condenser are grounded to the chassis with bolts. The engine bloack is grounded to the chassis with bolts. The points are grounded to the points plate, which grounds to the head, which grounds to the block. Depending on corrosion levels, you may need to service the ground connections.
The wiring on these bikes gives trouble over time with corrosion. The 3.5mm male and female connectors corrode over time and cause lots of problems. Even if a connector "looks good" you should dismantle it and use Caig's DeOxite and Q-Tips to clean both connectors. Smear a dab of silicon dielectric grease on these to prevent future corrosion, and you can eliminate the wiring as a possible problem.
The coils on these bikes hardly EVER fail. Keep in mind, these bikes require a battery-powered 6 Volt coil. An AC magneto coil from another Honda will not work. A 12 Volt battery-operated coil will also not work. You can run a CT90, S90, ar another ST90 coil. JC Whitney has a 6 Volt generic coil that works, and the 6 Volt Volkwagen Beetle coil works - but mounting it is tough to do.
The spark plug wires tend to erode up inside the wire where the plug boot threads on. Remove boot, trim the spark plug wire back unitl fresh metal wire shows, and thread the boot on. You can get a spark plug wire splicer kit if the spark plug wire is too far gone, since the wire is embedded in the ignition coil.
The spark plug boots fail OFTEN. Many had a resistor inside that burns out. Replace with a non-resistor boot only if you use no electronics near the bike.
As mentioned, the spark plug is problematic. I have seen more spark plugs for these engines defective right out of the box, than any other plug. Its the narrow construction perhaps.
If you think the grounds are bad, you can use jumper leads to connect the battery ground wire to the coil mount and cylinder head for a test.
The ignition switches sometimes fail, or develop resistance and resulting Voltage drop issues.
Hope something in here helps.