Seems potentially to be a lack of voltage/power issue, which could be down to battery condition, or battery connectors neither clean or tight
As described the battery has enough voltage to provide lights in the dash and to pull in the starter solenoid, which presents the battery with huge load requirements of the starter motor windings...which it can not cope with thus the voltage drops off rapidly, which in turn causes the solenoid to drop out of it`s attempt to run the starter..... and in so doing the voltage rises again, the dash lights illuminate once more..... and if the key/button is still held requesting the solenoid to re-energise the cycle will repeat repeat repeat...
