"Unable to cast string to double"?

Your problem is that game.StarterGui.ScreenGui.TextBox.Text is a string type, meaning it is text. SetMinutesAfterMidnight. An easy fix is to type tonumber(game.StarterGui.ScreenGui.TextBox.Text).

However, that is not your only problem

Another problem I encountered is this: game.StarterGui.ScreenGui.TextBox

When players edit the textbox in their Gui, it is edited in game.Players.LocalPlayer.PlayerGui, and not game.StarterGui, therefore, you will have to set your code to this:

script.Parent.MouseButton1Click:Connect(function() 
	print("clicked") 			 
	game.Lighting:SetMinutesAfterMidnight(tonumber(game.Players.LocalPlayer.PlayerGui.ScreenGui.TextBox.Text))
	print("entered") 
end)

You may be like "Wait a minute, this is super complex code"!

You can actually tidy this up with variables

local SetTimeButton = script.Parent
local Players = game:GetService("Players")
local Player = Players.LocalPlayer
local MinutesAfterMidnightInput = Player.PlayerGui.ScreenGui.TextBox
local Lighting = game:GetService("Lighting")

SetTimeButton.Activated:Connect(function()
	print("Clicked!")
	Lighting:SetMinutesAfterMidnight(tonumber(MinutesAfterMidnightInput.Text))
	print("Set time successfully!")
end)

Edit 1: Fixed error I just noticed