My passenger side headlight always seemed dimmer than the other one. It started to stop lighting up. I replaced the bulb this morning and now it will only stay on for a few minutes before turning off. The drivers side stays lit. I will be replacing the 12 volt battery tonight because it needs to be replaced. Would that cause this to happen?
No. Look for a problem with the light socket or wiring harness, for example the connection to body ground.
Assuming you have regular bulbs and not HIDs? If so, check the plug for the bulb. It may be melted slightly. Regardless, take the pins on the bulb and bend them inward very slightly, this will cause a stronger connection to form between the plug and the bulb pins. Also check for dirty or loose grounds (Places where wires simply bolt directly to the engine block or chassis)