As a first guess id say the video card/chip is going bad. it DOES happen, and their is dick you can do about that without replacing the mobo. However in case you are not totally fucked a few things to try that it might also be(that are fixable!).
1. while pc is on move the screen around on its hinges(very slowly up and down) if you have a wire with a short/that is loosing its connection it will turn off/flicker at certain points but be fine at others. If so you will know you need to (eventually) replace the wires/cut out the bad section(if the wire happens to have some extra length)
2. Bad caps/cracks in solder joints- bad caps are easy to tell look for bulging or leaking caps, if they all appear normal it could be micro fissures in the solder joints, you can fix that by reflowing the board with a heat gun or using an oven. Note that this will stink like ass and the fumes can be toxic so well ventilated area is a MUST. In case of bad caps simply desolder old ones and solder in new good ones.
3. It is possible but unlikely you are having a grounding issue, or a problem with a socket, as for any socket fixes it would be looking to a reflow to fix or replacing(or desoldering/resoldering completely), Since you have had to have several screens replaced it is quite possible the socket that holds the wires coming from your screen has been damaged/become loose. The best way to check that is to do a partial teardown of the laptop till you have access to the socket, turn the pc on and fiddle with the connector while its on(make sure you are grounded so as to not shock the mobo!) if the screen does crazy crap while your doing this then replace/fix it.
For just about ANYTHING on this the cost of bringing it to a tech(if they were even knowledgable enough to do any of this) would damn near outweigh the cost of getting a new laptop especially a lower end one. If it is a chip going bad on the mobo then this definitively true. Pretty much at the point you need to bring a PC to a tech its time to just chunk it and get a new one. Anyway those are the 4 most likely issues your experiencing, 3 of which are rather easy/cheap to fix, and only one of which is pointless. Though i should point out evne if it is the chip it COULD still be fixed with a reflow(potentially), however if the chip itself is going bad your just fucked. hope it helps