That is the headlamp night finder lamp fiberoptic cable. It's not a wire. It's sheathed in black. It slides into the back of the switch housing clear connector as you put it. There's a gray bulb holder about a foot down from there, maybe less. It comes apart and there's a special little bulb in there that is similar to a 194 bulb, but has a flat top on it for fiber optic. Can't recall the number. It's purpose was to shine a little blue light next to the headlamp switch to help you find it at night. It lit up with the ignition switch on. It's sort of useless, but it is what it is.
The only way those bozos could have broken the clip was if you had left the switch housing in the dash bucket when you sent the dash to them. You could pull those with small (8 mm head? maybe not?) screws. The fiber cable has a little brass sleeve that slides into the back of the clear clip.
It's interesting that I bought a new housing surround for the headlamp and dimmer switches years ago from the dealer, and it comes with a new clear clip on the back. Problem is, I don't know the part number and can't find it listed separately in the parts information. It appears it will have to be a junkyard gem item or get it from someone parting out a car. I'll continue to search and see if I can find a part number for the housing.
I would likely use an LED bulb 194 style. It should easily put out enough light to give you that blue hue you seek.
Here's a picture of the cable and gray bulb housing from another thread.
Here's one with the clear clip on the end.