Is there a way to utilize a old screen using electronics that can be handheld?
Laptop screens are usually custom manufactured for the likes of Dell and HP. Whilst they may use standard signals, you've got to figure out what they are and anything else in order to get it to work. Assuming the laptop is working, you can measure the voltages and observe the data to get an idea of what's what. Then you need to have hardware that can generate those same voltages and signals.
People have reverse engineered iPhone/iPod screens, so it can be done but it requires some skill and effort.
If you want a small system with a high resolution lcd, look at a tablet. You can find a myriad of no-name low cost Android tablets - I have a box of old tablets that I was given and not used. They are hi-tech waste :(
Learn how to write Android apps and you can have them do many useful things - even interface into Arduinos etc.