Been everywhere on Google, can't find this part name...I need the name of the whole box, not the fuse. The whole assembly. As you can see, the wire going from this box to the fuse box melted. Just need the part name. It's on a 2005 XG350 if that makes any difference.

 

I've been Googling for a while, been seeing it called a fuseable link or a fuse link...but then when I go to autozone for the part, it just shows the fuse, that's it...not the housing box or all the wires that it's connected to. I just need to know what the box is called and the wires that lead to the fuse box are called.

 

Are you asking about the power distribution box (the one with the fuses in it)?
Chrysler calls it Totally Integrated Power Module (TIPM)

It seems that all you need is a new cable and new terminal for the power distribution box.
Clean the terminal that was on the positive post of the battery.
Cut the old cable back to where the copper is clean.
Add a new cable by soldering new cable onto old with same size thickness.
Wrap well in electrical tape.
Connect a terminal to the end of the new cable.
Usually an eyelet that crimps onto the cable.
Some cables already come with them on

If you spoke with the sales person at the Auto parts stores
And/or show them the pictures then they should be able to help you.
Of course, buying a new cable, that has all the connections and what not,
will be much more expensive than buying the parts and making your own connection.

That part is called a battery cable
Part number 37200-39602
And the cost for it is about 126 dollars plus tax and shipping on Hyundaioemparts.com
Not suggesting you buy it there. Just saying that it where I quote the price

Personally, I would just cut that wire short and cover with electrical tape
Then buy a cable with eyelets on the ends.
Hook up one side to the battery cable where the nut is
And the other side to the power distribution box.

 

Yeah it appears to be a battery cable.