Sorry for reviving a v.old thread but have just gone through some troubleshooting with around 100 Nokia 5510 screens and this is a thread I found on the way.
Using the Adafruit 5510 library the top contrast is 127. If that doesn't give you a black rectangle the size of your screen, chances are you might need to reseat the LCD on the PCB. These are robust screens and unless there's a crack or you can see the liquid crystal leaking they should be able to be repaired fairly easily.
With the LCD powered on and connected take the top end (where it says Nokia5510) and squeeze the silver metal frame of the LCD towards the PCB gently - between thumb and forefinger. You should see the contrast change (it should darken). If that works it may benefit you to bend the metal pins that attach the frame of the LCD to PCB slightly to achieve a better contrast. Then adjust it back down in code.
If that's not worked there's one last thing to try. Remove the LCD from the PCB by compressing the metal pins that attach the frame of the LCD to the PCB. Once you have two pieces remove the rubber bung connector that's sat in the back of the LCD then reseat it gently. Reassemble and try again.
Note: Every time you remove the LCD to do something, reset the Arduino board or it won't refresh your changes and will stay blank.
Hope this helps someone!