Solenoid rattling is nearly always due to low voltage at the high-current solenoid terminal. This is likely to be caused by low battery capability, a poor connection at one of the battery terminals, or a poor connection at a solenoid terminal. Measure the voltage at the solenoid's high current terminal, and see what happens to it when you operate the starter. Usually the solenoid chatters if it drops below about 5 Volts. If it chatters with a good voltage level under load (say more than 8-9 Volts) either the voltage is low at the solenoid's activation terminal due to faulty wiring or starter switch, or the solenoid is faulty. Faulty solenoid is the least common of the possibile causes.
It is very common on elderly ride-on mowers to find that the wiring has been raped and pillaged by a previous owner. We usually describe this type of alteration as "tenant's improvements" by a previous owner. The best thing to do in such cases is to get a wiring diagram and restore the wiring to its original state. The alternative is to fix each "improvement" in turn as the malfunctions it is causing become evident. Hence, once you have checked the battery capacity and the high current connections as described above, I suggest you take a general look for dodgy wiring.
