How can I identify the side of an object the mouse's click was?

You can use the math function math.abs() to construct a new Vector3 from the absolute value of each components of the p vector and then retrieve it's X component:

local tool = script.Parent


local function onButton1Down(mouse)
    local part = mouse.Target

    -- Just to make sure the mouse isn't pointing nowhere or at the baseplate
    if part and part.Locked == false then
        local vec = mouse.Hit:toObjectSpace(part.CFrame).p

        print(Vector3.new(math.abs(vec.x), math.abs(vec.y), math.abs(vec.z)).X)
    end
end

local function onEquipped(mouse)
    mouse.Button1Down:connect(function() onButton1Down(mouse) end)
end


tool.Equipped:connect(onEquipped)

Hope this helps. If it did, please leave an upvote and mark my question as answered. :) If you need anything else, leave me a message.