2000 Mitsubishi Eclipse won't start cold

From the description, I'd suspect either the alternator, a defective starter solenoid, a defective starter relay, or corroded battery terminal contacts.

I'd start with the cheapest/easiest first and thoroughly clean both of the cable connectors at the battery using a stiff wire brush. I'd also check and clean/tighten the connections to the chassis ground and at he starter solenoid itself. If these fail to solve your problem, then I'd make sure the alternator is charging the battery since even a brand new battery would need to get charged at some point so would be drained. (Check the voltage at the battery when the engine is off and again when running to make sure it is getting charged.) If none of these solve the problem, I'd look at the starter solenoid. If it is the solenoid, you can generally get the vehicle to crank over by tapping lightly on the solenoid with a small hammer. (The vibrations usually cause the solenoid contacts to close properly). When and if you go that route, then you need to have an assistant to turn the key and need to exercise a great deal of caution with your self since the engine can start at any time. If this starts the vehicle, then replace the solenoid, or if it comes on the starter as an assembly, replace that.

Good luck

Bob

Best guess would be the contacts in the starter solenoid. Good info here...