ClickDetector for one Player only?

Use a LocalScript with FilteringEnabled (game.Workspace.FilteringEnabled = true) and have the LocalScript add a ClickDetector to a part and connect a clicked event to it.

-- LocalScript
local player = game.Players.LocalPlayer
local playerAllowed = "YourName"

if player.Name ~= playerAllowed then return end

local click = Instance.new("ClickDetector")
local part = game.Workspace:WaitForChild("ClickPart")
click.Parent = part

click.MouseClick:connect(function()
    print("Clicked by", player, ";", script) -- This is debugging so you know it did for sure come from that player.

    -- Your code here
end)