How can bring an iPhone 4 back from the dead?

Start off with Replacing the battery and reconnecting and cleaning the power flex connection with appropriate alcohol solution would help - Then Try Charging via Wall Socket and Computer USB

After you can get it to stay on test a few of the parts such as : Cameras, Vibrator, Volume Buttons, Earpiece Speaker and Loudspeaker.

The "Logic board" IS your phone. The only way to replace it is with another phone. The only thin that was could be irrepairably dammaged is the screen. everything else can be disassembled, cleaned with alcohol, sprayed with brake-kleen, then blown dry with compressed air. The only phones(actually its the "logic board" that determines success here) I havent been able to recover by this method are those that 1)Got wet, 2)Were allowed to corrode, and 3)Suffered physical damage due to components being eaten away so that after cleaning they were simply not there any more. Get a jewlers loop and look at the size of the smallest components on the logic board. They look very important and you can fit at least 3 on the head of a pin.