Sorry to hear of your problems.
Three types of R51 graphics - ATI mobility Radeon 7500, Radeon 9000 and Intel extreme graphics 2.
If your R51 has the Intel graphics chip, then it's very unlikely you'll have a graphics chip fault.
However you may well have the fault with either of the ATI chips.
A quick look with the keyboard removed will reveal the the graphics chip and manufacturer.
To test whether you have a graphics fault, connect to an external monitor and see if that works.
Also try
gently flexing the TP and see if the display comes on.
Ditto with pressing down on the graphics chip.
Note that depending on the bios settings, the ext monitor may not work (not sure why this is, I've found it to be so on a number of them through my hands).
Remove ac adapter, cmos and internal battery. Short out the cmos battery pins for 10 secs.
Still with no cmos battery, power up on a.c. (you'll be doing this 'blind'!) and after a few secs press F1 to enter bios.
Wait for 1 minute to make sure it has entered bios, press F9 to reset the bios to default (to make sure even though it should have done with the cmos batt out), and F10 to save config and restart.
It should now display in the ext monitor if there's no graphics fault.
The graphics chips have approx 669 tiny 'pins' (yeah sad - counted every one of them!!) under a roughly 30mm sq chip. These pins are known as a Ball Grid Array or BGA and actually consists of tiny solder balls:
There are loads of articles but read this for more information
http://www.siliconfareast.com/bga.htm
What happens is that due to poor chip design, there are localised hot spots within the chip which causes the chip to warp and can cause some of the BGA balls to fracture from the motherboard.
The fault then appears when the TP's flexed or simply that the balls are no longer touching the main board.
The phrase 'reflowing' means to reheat the chip to melt the bga and hopefully cause the balls to be resoldered back onto the main board. Resoldering in this context means the same.
Have a look at the sticky in this section about how to do it:
http://forum.thinkpads.com/viewtopic.php?t=57021
This is a typical 'home brew' method which many people have successfully done themselves.
Also do a search for BGA reflowing on Youtube, since there are some great demos on how to do it.
If there's any sort of official warranty with your R51 then it needs to be returned to be sorted.
Also whilst a good number of people have sucessfully used this method, there are many others who have permanently burnt out their boards.
There are others here, typically
jamiphar who offers a first class reballing and reflowing service for graphics chips. I did read recently though that he's inundated with work and may not be taking on any more for a while.
Reballing means removing all the individual tiny solder balls and renewing them before reflowing the chip BGA onto the motherboard.
Bottom line - thoroughly research the subject within these forums and elsewhere.
it's only then that you can decide as to whether you feel able to do it yourself.
HarryC points out very wisely that it's not for the feint hearted and there's considerable risk with permanent motherboard damage.
Let us know how you get on and get back with any questions you have etc.
Good luck!