[SOLVED] How do make it when I hoverover a Part that has BoolValue UI appears and follows the mouse?

Why did you put UserInputService in this? We are needing the user's mouse, not keyboard clicks. There is also no need for the player added event since we know that local scripts run for the current player only. At least I hope you used a LocalScript which you must

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()  


if mouse.Target then
    if mouse.Target:FindFirstChild("BoolValue") then
        player.PlayerGui.CollectGUI.Enabled = true --ScreenGUI is enabled
        player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, mouse.X, 0, mouse.Y) --Frame follows mouse
        player.PlayerGui.CollectGUI.Frame.ItemName.Text = mouse.Target.Parent.Name -- ItemName.Text is the part's name
    else
        player.PlayerGui.CollectGUI.Enabled = false --ScreenGUI is disabled
        player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, 0, 0, 0) --Frame position is reset
        player.PlayerGui.CollectGUI.Frame.ItemName.Text = "" --ItemName.Text is reset
    end
end

So you may wonder why that script is not working either.

Well you need a loop which constantly checks the mouse's Target.

local RunService = game:GetService("RunService")
local player = game.Players.LocalPlayer
local mouse = player:GetMouse()  

RunService.Stepped:Connect(function(step)
    --wait(step) --Optional if you want it to coordinate with user's frames
    if mouse.Target then
        if mouse.Target:FindFirstChild("BoolValue") then
            player.PlayerGui.CollectGUI.Enabled = true --ScreenGUI is enabled
            player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, mouse.X, 0, mouse.Y) --Frame follows mouse
            player.PlayerGui.CollectGUI.Frame.ItemName.Text = mouse.Target.Parent.Name -- ItemName.Text is the part's name
        else
            player.PlayerGui.CollectGUI.Enabled = false --ScreenGUI is disabled
            player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, 0, 0, 0) --Frame position is reset
            player.PlayerGui.CollectGUI.Frame.ItemName.Text = "" --ItemName.Text is reset
        end
    end
end)

I wish you goodluck on solving your issue if my answer did not help.

just remove the event called PlayerAdded. Local scripts can't seem to use that event. the following code is the fix.

local Players = game:GetService("Players")

local UIS = game:GetService("UserInputService")

     UIS.InputChanged:Connect(function(input)

        local player = Players.LocalPlayer
        local mouse = player:GetMouse()  
    if mouse.Target then
        if mouse.Target:FindFirstChild("BoolValue") then
            player.PlayerGui.CollectGUI.Enabled = true --ScreenGUI is enabled
            player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, mouse.X+3, 0, mouse.Y+3) --Frame follows mouse
            player.PlayerGui.CollectGUI.Frame.ItemName.Text = mouse.Target.Parent.Name -- ItemName.Text is the part's name
        else
            player.PlayerGui.CollectGUI.Enabled = false --ScreenGUI is disabled
            player.PlayerGui.CollectGUI.Frame.Position = UDim2.new(0, 0, 0, 0) --Frame position is reset
            player.PlayerGui.CollectGUI.Frame.ItemName.Text = "" --ItemName.Text is reset
        end
    end
end)

Since we're inside a player via the local script its unnecessary to call that event for all players on the server. Now if this was a server side script then it would work!.

just incase your wondering on line 1 the:

local Players = game:GetService("Players")

is the same as:

local player = game.Players.LocalPlayer