It’s just a 5mm white LED, one of about two dozen in an array. Can I just get a replacement off eBay or is it more complicated than that. I do have a multi-meter if I need to measure something to find one with the right specs.
LEDs come in two basic varieties, those with the driver circuitry built into them and those without. The ones with the driver circuitry are much less common. Most of the time you find just the LED. If yours are part of a strand, it’s pretty much guaranteed that it’s just the LED and doesn’t have the driver built into it. Make sure you buy the same type.
You need to make sure that the LED you buy can handle the current going through the strand. The brightness level of the LED may not exactly match. Since they are in an array, I’m not sure if that will be an issue for you or not.
If you don’t know the current going through your LED strand, you can pull out a known working LED and attach it to a power source with a variable resistor. Make sure that the variable resistor is at its highest value when you turn on the power, then lower the resistance until the LED gets bright. That will give you a rough idea of the current requirements of that LED type.
If you really need to match brightness and size and things like that, you might need to contact the manufacturer of whatever this is and get the manufacturer and part number for the LED that they used.
There’s a wide variety of 5 mm white LEDs out there, with varying efficiencies, IV curves, wavelength ranges, viewing angles, etc.
Measuring the voltage drop and current through the LED would allow you to find a replacement LED that would likely be safe to use from a strictly electrical perspective. But if you’re wanting it to have the same *optical *characteristics as the other LEDs, i.e. you want it to “look the same” as the other LEDs, it will take more detective work.
I’m not sure about that. Measuring the voltage drop won’t tell you what maximum current the LED can handle.
Well I also mentioned the current through the LED should be measured. 
He would then pick an LED that can safely handle that magnitude of current (with a healthy safety margin thrown in, of course).
If only one LED is out, it is in parallel.
Jameco has just about any LED made. What device is this array? Can you get at the LED to remove it or measure voltages?
Maybe its the current limitter for that LED that is dead.
test to see if there is suitable voltage across the diode…
Probably the current limitter is built in, so then you replace the diode.