@Zoredache is correct.
It's called a POST card and they have an LED display. The display shows the value that is written to port 0x80 (or possibly other ports, not sure if that's a standard). BIOSes write to this port as they progress through their startup sequence. If something causes the sequence to stop, the last written value can reveal why.
Some POST cards have more than one edge connector (i.e. ISA and PCI connectors, the card is rotated according to what system you want to connect it to) and I did see in searching mini-PCI and mini-PCIe for laptops.
Of course this is specific to each BIOS manufacturer and I'm not sure at all how UEFI handles this.
There also isn't a lot of user-replaceable hardware on modern boards unless you have things like a rework station, etc. Maybe you do, though. :)