I’m making a cloud system which orbits around a planet.

I made it so it orbits the planet in a slow trajectory, however I want it to spawn randomly anywhere because its starting position is only at one unknown point, which I have no idea on how to achieve this.

You could add a XOffset and a YOffset in math.cos(angle)
it would look a bit like math.cos(angle+XOffset)...

The XOffset and YOffset could be randomized via math.random()*360 or predetermined:

local predeterminedXOffsets = {--list of possible xoffsets}
local predeterminedYOffsets = {--list of possible yoffsets}
local XOffset = predeterminedXOffsets[math.random(1,#predeterminedXOffsets]
local YOffset = predeterminedYOffsets[math.random(1,#predeterminedYOffsets]
...

or if you want to use randomized completely:

local XOffset = math.random()*360
local YOffset = math.random()*360
...