How would you use a mouse to select players when clicked?

It depends what you mean by "Select a player."

To start out, if I were you I would probably write a script to insert a ClickDetector to each of the Player's limbs when they join the game, then write a script that detects when the ClickDetector is clicked.

function assignClick(Player)
ClickDetector = Instance.new("ClickDetector")
p = game.Workspace.Player:GetChildren()

for i,v in pairs(p) do
    if v:IsA("Part") then
    ClickDetector.Parent = v
    else
    return
    end
end
end

game.Players.PlayerAdded:connect(assignClick)

That script should assign ClickDetectors in each of the Player's limbs.

noooo i would not use click detectors. I would use the mouse in a LOCAL script.

local p = game.Players.LocalPlayer
local m = p:GetMouse()

function showprofile(player) -- change this to a gui, or whatever
print(player.AccountAge)
print(player.userId)
print(player.Name)
end

m.Button1Down:connect(function()
local target = m.Target
if target and target.Parent and m then
if game.Players:GetPlayerFromCharacter(target.Parent) then
showprofile(game.Players:GetPlayerFromCharacter(target.Parent))
end
end
end)

using the mouse is a lot easier and cleaner than using a click detector.