I stuffed up an already broken (but working with a bodge) mainboard last night, so I removed the LCD to sell on eBay, but as I was asked how to remove an H1xx LCD without damaging anything, here goes: (all the pictures are clickable for larger 3x size)
DISCLAIMER: You undertake any modification/repairs on your own, I will not be held responsible for your actions.
1. Strip the player down to just the mainboard with no battery attached:

(screen is on just to show it is working before removal, mainboard is broken beyond repair though)
2. Peel back the tape covering the LED backlight:
3. Very carefully wedge a craft knife inbetween the backlight strip and the mainboard and unsolder the two points marked (don't try and remove the backlight strip from the LCD), using the knife to gently lever the backlight strip away from the mainboard during the unsoldering:
4. Once the backlight has been unsoldered from the mainboard take a look at the back of the mainboard and you'll see 4 points, these are how the LCD holds in place on the mainboard, two points which just sit in and two clips:
5. Very carefully lever the LCD off the mainboard at the "point" corners, it may be a little difficult to lever off because of the tape round the LCD (don't remove it completely) but be gentle:
6. This is just to show the LCD still works after being lifted off the mainboard, the ribbon cable underneath is still connected:
7. Finally carefully lift back the LCD screen, pull back the brown holding flap of the connector socket as indicated, then carefully peel the ribbon cable off the mainboard (it's held down by a little double sided sticky tape) and out of the socket:
9. Et voila, one removed LCD screen, once you've successfully removed one you'll find replacing it is just as easy.
I'll be listing this LCD screen along with a H120 shell on eBay very soon.