Why is my namehide GUI not working right?

The property HealthDisplayDistance and NameDisplayDistance change how far you can see other player's names, not how far other player's see your name.

The only way I know how to do this is an old hacky method where you make your player's head invisible and put a fake head in it's place.

local player = game.Players.LocalPlayer
local nameVisible = true

function HideName()
    if player.Character then
        local fakeHead = player.Character.Head:Clone()
        local weld = Instance.new("ManualWeld")
        fakeHead.Name = "FakeHead"
        fakeHead.Parent = player.Character
        weld.Part0 = player.Character.Head
        weld.Part1 = player.Character.FakeHead
        weld.Parent = player.Character.FakeHead
        player.Character.Head.Transparency = 1
    end
end

function ShowName()
    if player.Character and player.Character.FakeHead then
        player.Character.FakeHead:Destroy()
        player.Character.Head.Transparency = 0
    end
end

textbutton.MouseButton1Down:connect(function()
    if nameVisible then
        HideName()
        textbutton.BackgroundColor3 = Color3.new(255, 45/255, 0/255)
        textbutton.Text = "Name Hidden"
        nameVisible = false
    else
        ShowName()
        textbutton.BackgroundColor3 = Color3.new(100/255, 255/255, 50/255)
        textbutton.Text = "Hide Name"
        nameVisible = true
    end
end)

Value = Instance.new("BoolValue")
    Value.Parent = script.Parent
    Value.Value = false
---If it's set to LocalPlayer it will display everyones health and name to 0.
---Because that property is like the only one that can set everyone's to zero.
---I suggest creating a event and use player
---Here lets make one
game.Players.PlayerAdded:connect(function(player)
textbutton.MouseButton1Down:connect(function()
    if Value.Value == false then 
       player.HealthDisplayDistance = 0    
       player.NameDisplayDistance = 0
        textbutton.BackgroundColor3 = Color3.new(255, 45/255, 0/255)
        textbutton.Text = "Name Hidden"
        Value.Value = true
    elseif Value.Value == true then
       player.HealthDisplayDistance = 100
        player.NameDisplayDistance = 100
        textbutton.BackgroundColor3 = Color3.new(100/255, 255/255, 50/255)
        textbutton.Text = "Hide Name"
        Value.Value = false
    end
end)    
end)
--Accept this answer if it helped I simply changed client to server sided. So it does not confuse the property with client-sided.

What you're doing is trying to set their name/healthviewdistance to 0, which simply makes it so that the player can't see any names at all.

What you want to do instead is set their head's transparency to 0 and make a "fake head" so that their nametag disappears. This is because if a player's head is transparent then their name does not show up. However, this means that it would look like the player doesn't have a head! To fix this, we just clone the head and weld it.

Here's the code you'll want -

local player = game.Players.LocalPlayer
local hidden = false-- this is the bool I used to check if your name is hidden
script.Parent.MouseButton1Down:connect(function()
if hidden == false then
    local fakeHead = player.Character.Head:Clone() --you're cloning the head here
    fakeHead.Name = "FakeHead"
    fakeHead.Parent = player.Character
    player.Character.Head.Transparency = 1 -- this is where you hide the name.
    player.Character:WaitForChild("FakeHead")
    local weld = Instance.new("Weld") -- welding it so it doesn't fall off :)
    weld.Parent = player.Character.Head
    weld.Part0 = player.Character.Head
    weld.Part1 = player.Character.FakeHead
    script.Parent.TextStrokeColor3 = Color3.new(170/255,0/255,0/255)
    script.Parent.Text = "Show name" -- giving feedback
    hidden = true
else if hidden == true then
    player.Character.Head.Transparency = 0 -- showing the name
    player.Character:FindFirstChild("FakeHead"):Destroy() -- destroying the fake head
    script.Parent.TextStrokeColor3 = Color3.new(156/255,156/255,156/255)
    script.Parent.Text = "Hide name" -- giving feedback
    hidden = false

end
end
end)