I wish to refit starter button or a new Start switch, has battery, starter motor,solinoid,Did have button before it was given to me, how to wire it is the question, where to on solinoid sw, there is one terminal with no wireing going to it?

If there is only one terminal then it goees to that one and the other wire on the "button/switch" should go to battery positive. You might want to put a fuse or circutbreaker inline on the positive side in case you ever get a short.

There is two terminals on the solinoid but only one is being used, it has ignition wiring batteryand wiring to starter, maybe ignition is stuffed, as has one terminal with no wire.

I have seen some solenoids with 2 terminals, may be the type that needs a ground for the second terminal.
That groud terminal may even go through some safetys, Neutral, Clutch Safety etc.
See if applying voltage to the terminal energizes, if not, apply a ground to the second and try again, fused would be a safe way to test. Not sure what this is(Lawnmower?), Be careful not to test without being ready for it to turnover. Something's can injure or kill.
Good Luck