2001 Oldsmobile Aurora Daughter's Car

I am having trouble starting this car about 50% of the time. When it doesn't start - I turn the key to the start position and all of the lights (dome, dash) go out and all I hear is a click sound from the engine area. Once I let off of the key all of the power is restored, lights come back on. I have replaced the battery, but no help. Then the car may start fine for 2 or 3 days in row with no issues. Do not want to spend a great deal of money on this car troubleshooting issues.