Resolved
I thought I would check back in and let you all know that I finally found the problem. It was a ground and inverter issue. The client that owns this laptop had already purchased a new screen and inverter before bringing it to me.
Not thinking a new inverter would be bad and trying the old inverter as well, I decided to use what I know from working with electrical issues in automobiles and knew it had to be a ground related issue.
Taking out all the parts the client had purchased and replacing them with the original parts, the laptop screen was still dim. I pulled out the inverter and examined it closely and noticed that the grounding pad was being covered by the clear plastic and could not get a good connection, so I partially removed a small piece of it to expose the grounding pad. I reconnected the inverter, powered the unit on, and the screen lit up! However, after booting into the OS, the screen went dim after about 3 minutes. With the unit still on, I began slowly turning the small grounding screw holding the inverter in place and the screen flickered on and off. I powered the unit off, ran a separate ground wire and booted the system again to find out the same thing happened: after about 3 minutes the back light could go dim and If I turned the grounding screw it would come on again and go back off.
Not having much to do yesterday and loving to figure problems out, I decided to do some experimenting with what I knew was a bad inverter, as the client had originally expected. They just happened to get another bad inverter when they ordered a replacement.
Knowing the inverter would work if the ground screw was turned and that this was a ground issue, I knew it had to be internal to the inverter because I had already ran ground straps and they wouldn't work. So, I broke out my hot air station (which actually worked this time) and began re-flowing the inverter. After allowing it to cool for a few minutes, I reconnected it back to the laptop, booted it up, and allowed it to sit for a few minutes. It actually stayed on for over 3 minutes. The laptop has been on for almost 24 hours now and hasn't went dim yet. So, I've already got a new inverter on the way.