How do I make a gui script when hovering over an object with a mouse?

PlayerMouse

For something as simple as having a name tag appear near the mouse when you hover over an object, you shouldn't need to use RayCasting at all. In fact, this can be accomplished with the Move event and Target property of the PlayerMouse.

Explanation

The goal is to make it so whenever the mouse moves, it will check to see what object it's hovering over, then do whatever it is we want to do. Here's an example:

local Player = game:GetService("Players").LocalPlayer -- Get the player
local Mouse = Player:GetMouse() -- Get the player's mouse

-- Just creating a name tag for this example. This is assuming the script is inside a ScreenGui that's in the PlayerGui.
local Tag = Instance.new("TextLabel",script.Parent)
Tag.Text = ""
Tag.Visible = false
Tag.TextColor3 = Color3.new(1,1,1)
Tag.Font = "ArialBold"
Tag.FontSize = "Size18"
Tag.BackgroundColor3 = Color3.new(0,0,0)
Tag.BackgroundTransparency = 0.5
Tag.TextXAlignment = "Left"

-- Connect the "Move" event to the mouse 
Mouse.Move:connect(function()
    local Target = Mouse.Target -- Get the object the mouse is hovering over
    if Target then -- Make sure the target exists

        -- Update the information when a target exists
        Tag.Text = Target.Name
        Tag.Size = UDim2.new(0,Tag.TextBounds.X,0,20) -- TextBounds.X will make the tag be as long as the text, giving it a more complimenting appearance
        Tag.Visible = true

    -- If no target exists, then...
    else

        -- Hide the tag
        Tag.Visible = false

    end 

    -- Update the tag's position with the mouse
    Tag.Position = UDim2.new(0,Mouse.X-Tag.AbsoluteSize.X,0,Mouse.Y-Tag.AbsoluteSize.Y)
end)

Result should look like this: https://gyazo.com/25cca50513b0137af95a3169d42445f0 (remembering this should be in a local script, inside a ScreenGui object).

Edit

As XAXA mentioned, obviously this won't work unless you actually move the mouse. Unlike if you were to walk by something with your mouse in the same place, the code will not update. We could easily fix this by using the RenderStepped update of RunService to make it so our code will constantly check it's conditions regardless of user input. Here's an example:

local RunService = game:GetService("RunService")
local Player = game:GetService("Players").LocalPlayer -- Get the player
local Mouse = Player:GetMouse() -- Get the player's mouse

-- Just creating a name tag for this example. This is assuming the script is inside a ScreenGui that's in the PlayerGui.
local Tag = Instance.new("TextLabel",script.Parent)
Tag.Text = ""
Tag.Visible = false
Tag.TextColor3 = Color3.new(1,1,1)
Tag.Font = "ArialBold"
Tag.FontSize = "Size18"
Tag.BackgroundColor3 = Color3.new(0,0,0)
Tag.BackgroundTransparency = 0.5
Tag.TextXAlignment = "Left"

-- Using the RenderStepped event of RunService, which updates at roughly 1/60th of a second (about twice as fast as the generic wait function, obviously this should still be in a local script)

RunService.RenderStepped:connect(function()
    local Target = Mouse.Target -- Get the object the mouse is hovering over
    if Target then -- Make sure the target exists

        -- Update the information when a target exists
        Tag.Text = Target.Name
        Tag.Size = UDim2.new(0,Tag.TextBounds.X,0,20) -- TextBounds.X will make the tag be as long as the text, giving it a more complimenting appearance
        Tag.Visible = true

    -- If no target exists, then...
    else

        -- Hide the tag
        Tag.Visible = false

    end 

    -- Update the tag's position with the mouse
    Tag.Position = UDim2.new(0,Mouse.X-Tag.AbsoluteSize.X,0,Mouse.Y-Tag.AbsoluteSize.Y)
end)

And that's how we'd fix that problem. Now regarding your question about making it only happen for certain parts, we could create a dictionary of parts with a true value to easily index and compare it to our Target property. Here would be the final result:

local RunService = game:GetService("RunService")
local Player = game:GetService("Players").LocalPlayer -- Get the player
local Mouse = Player:GetMouse() -- Get the player's mouse

-- Dictionary of parts that have a true (or existing) value
local Objects = {
    [workspace:WaitForChild("Part1")] = true,
    [workspace:WaitForChild("Part2")] = true,
    [workspace:WaitForChild("Part3")] = true
}

-- Just creating a name tag for this example. This is assuming the script is inside a ScreenGui that's in the PlayerGui.
local Tag = Instance.new("TextLabel",script.Parent)
Tag.Text = ""
Tag.Visible = false
Tag.TextColor3 = Color3.new(1,1,1)
Tag.Font = "ArialBold"
Tag.FontSize = "Size18"
Tag.BackgroundColor3 = Color3.new(0,0,0)
Tag.BackgroundTransparency = 0.5
Tag.TextXAlignment = "Left"

-- Using the RenderStepped event of RunService, which updates at roughly 1/60th of a second (about twice as fast as the generic wait function, obviously this should still be in a local script)

RunService.RenderStepped:connect(function()
    local Target = Mouse.Target -- Get the object the mouse is hovering over
    if Target and Objects[Target] then -- Make sure the target exists, and that it's inside the exclusive Objects table

        -- Update the information when a target exists
        Tag.Text = Target.Name
        Tag.Size = UDim2.new(0,Tag.TextBounds.X,0,20) -- TextBounds.X will make the tag be as long as the text, giving it a more complimenting appearance
        Tag.Visible = true

    -- If no target exists, then...
    else

        -- Hide the tag
        Tag.Visible = false

    end 

    -- Update the tag's position with the mouse
    Tag.Position = UDim2.new(0,Mouse.X-Tag.AbsoluteSize.X,0,Mouse.Y-Tag.AbsoluteSize.Y)
end)

Edit 2

For your third question, making it so you don't have to manually assign the objects and to just make it so all objects in a model are active, you could use the IsAncestorOf method on the model, with the Target property.

local RunService = game:GetService("RunService")
local Player = game:GetService("Players").LocalPlayer -- Get the player
local Mouse = Player:GetMouse() -- Get the player's mouse

-- Get the model we'll need to hold all our active objects
local ActiveParts = workspace:WaitForChild("SomeModel")

-- Just creating a name tag for this example. This is assuming the script is inside a ScreenGui that's in the PlayerGui.
local Tag = Instance.new("TextLabel",script.Parent)
Tag.Text = ""
Tag.Visible = false
Tag.TextColor3 = Color3.new(1,1,1)
Tag.Font = "ArialBold"
Tag.FontSize = "Size18"
Tag.BackgroundColor3 = Color3.new(0,0,0)
Tag.BackgroundTransparency = 0.5
Tag.TextXAlignment = "Left"

-- Using the RenderStepped event of RunService, which updates at roughly 1/60th of a second (about twice as fast as the generic wait function, obviously this should still be in a local script)

RunService.RenderStepped:connect(function()
    local Target = Mouse.Target -- Get the object the mouse is hovering over
    if Target and ActiveParts:IsAncestorOf(Target) then -- Check if the part is anywhere inside the model

        -- Update the information when a target exists
        Tag.Text = Target.Name
        Tag.Size = UDim2.new(0,Tag.TextBounds.X,0,20) -- TextBounds.X will make the tag be as long as the text, giving it a more complimenting appearance
        Tag.Visible = true

    -- If no target exists, then...
    else

        -- Hide the tag
        Tag.Visible = false

    end 

    -- Update the tag's position with the mouse
    Tag.Position = UDim2.new(0,Mouse.X-Tag.AbsoluteSize.X,0,Mouse.Y-Tag.AbsoluteSize.Y)
end)

Hope that helped. If you have any questions, just let me know.