Making a ScreenGUI Dialog Textbox?

The main problem here is that you are using StarterGui. It is not where the players store their currently-viewing GUIs - it is a place where GUIs get cloned from to be put into the players' PlayerGui.

You will find that if you respawn, you should see the dialog box.

As well as that, the GUI part is simply not a job for a server-side Script. Instead, use two different scripts - a LocalScript and a normal Script.

Normal Script (AKA server-side Script):

local Gui = script.DialogGui -- put the ScreenGui inside the server-side script

script.Parent.ClickDetector.MouseClick:Connect(function(PlayerWhoClicked)
    local Clone = Gui:Clone() -- makes a copy of the gui
    Clone.Parent = PlayerWhoClicked.PlayerGui -- 'pastes' it into the player's PlayerGui
end)

LocalScript:

local Gui = script.Parent -- say this was a LocalScript inside the ScreenGui
local Dialog = Gui.Dialog
local PersonsName = Dialog.PersonsName
local Player = game.Players.LocalPlayer -- only works in LocalScripts

-- don't need to wait for any events since the script runs after it gets cloned by the server-side script
PersonsName.Text = Player.Name
-- you should not need to make PersonsName visible since setting the visibility of something holding other things also makes the things inside it the same visibility
Dialog.Visible = true
-- you do not really need to make it toggle visibility for something this simple, since it is meant to be visible when it comes into existence anyway

-- closes the dialog after 5 seconds
wait(5)
Gui:Destroy()

Hope I helped!

~TDP

Please read the community guidelines before you make a question next time. This post applies to #4 in General Website Rules, which says that ScriptingHelpers are a help site, not a request site. Provide some sort of evidence that you tried to do this yourself and aren't asking for it to be done for you and I'm confident that many people would be more than happy to help you.

Use this in a local script

debounce = false
function OnClicked()
    local Dialog = game.Players.LocalPlayer.PlayerGUI.Dialog.Dialog
    Dialog.Visible = true
    local DialogName = game.Players.LocalPlayer.PlayerGUI.Dialog.PersonsName
    DialogName.Visible = true
end

script.Parent.ClickDetector.MouseClick:connect(OnClicked)