How would you connect a LVDS display to a laptop motherboard supporting natively a different display?
The LVDS interface standard should be pretty straight-forward, but it is only that - an interface standard. It only describes, which (and how many, 2 or 4) pairs of wires carry signals from the controller to the LCD panel, and at what voltages. But it does not say what the signal means to the controller or the panel. This is done by the display driver, and is specific to the panel in use (is it 16-bit, or 18-bit, or 24-bit color display; front and back porch requirements for horizontal and vertical scan lines; refresh clock rates and display resolution; etc.). The display driver may implemented be in hardware or in software.

On the other hand, the connection interface does not necessarily mean you are stuck with a specific controller. After all, even an Arduino can produce the signal to drive a VGA display, so you can definitely could find or create a solution to drive your LVDS display... It depends on the board you want to use for your machine, but you have provided no specific information at all.

My final advice is to find the make and model of your LCD panel and look on Ebay for a driver board. Those are to be found for most LCD panels produced in the last 10-15 years, and cost in the range of $ 15 to $70 USD. So find a driver board that fits your LCD and experiment with it. Usually they offer different input signals - RCA, S-Video, VGA, DVI, HDMI, or all of them together. And the LVDS is then how the driver board connects to the display. After you have played around with it a bit, you can see if you can use this same board in your machine (if you remove the unneeded connectors) or just create your own  driver on the basis of the first board by using the same parts, but making it fit the format of your new machine....