How to capture values displayed on 16x2 LCD
This would be simple, as you are essentially going to have to create the LCD driver's control interface in code. Assuming an 8 bit interface, you need between 10 or 11 pins (If R/W is used instead of tied as Write mode). If 4 bit, then you need 4 less. As the HD44780 LCD protocol is fairly well known, it would only be a matter of listening to the data pins when E is toggled, then parsing the data. Data is ASCII, while the commands are simple to understand. At that point you can log it to an SD card, or send it via serial or whatever.

The biggest issue will be timing. Not knowing how often the mcu writes to the lcd, or how fast, timing will be an issue. Not to mention, any Arduino library overhead. DigitalRead() will take much longer than direct reads, so it's suggested that you code it directly. A quick google shows no one else has produced a HD44780 sniffer or interceptor. This is essentially a logic analyzer. Keep in mind you could simply read and throw the data to a computer to process, as that may be faster.

Oh, and if you do this, be sure to share it online and blogs and stuff. A HD44780 sniffer like that is sure to be popular.