How would I make a pairs loop go in a random order?

If you want to choose random indicies from an array, generally the best solution is to randomize the array before the iteration. There are many algorithms for randomizing, or "shuffling" arrays, the most popular probably being the Fisher–Yates shuffle.

Here's a Lua implementation:

local function randomize(t)
	local _t = {}

	-- make a copy of the original table
	for i = 1, #t do 
		_t [i] = _t [i]
	end

	-- shuffle the copied table randomly
	for i = #_t, 2, -1 do
		local j = math.random(i)
		_t[i], _t[j] = _t[j], _t[i]
	end

	return _t
end

Now, you just have to pass your array through this function then iterate over the result as you would normally. For example:

for _, v in next, randomize(workspace:GetChildren()) do
	print(v)
end

Let me know if you have any questions.