How to make the value of 'BoolValue' remain the same after player dies?

Answer: Use remote functions and server storage. 1. Put a remote function in Replicated Storage so client and server can access it. 2. Add a folder of the player's x2 speed values. 3. Add a BoolValue of the player's name in the folder when they join using a server script in ServerScriptService.

local speedFolder = Instance.new("Folder")
speedFolder.Name = "DoubledSpeed"
speedFolder.Parent = game.ServerStorage

game.Players.PlayerAdded:Connect(function(player)
    local plrBoolValue = Instance.new("BoolValue")
    plrBoolValue.Name = player.Name
    plrBoolValue.Value = false
    plrBoolValue.Parent = speedFolder
end)

1. Do a check function for checking the BoolValue in the folder. Needs to be a server script again.

-- IF SEPARATE SERVER SCRIPT FROM ABOVE SCRIPT:
local speedFolder = game.ServerStorage:WaitForChild("SpeedFolder")
local remoteFunction = game.ReplicatedStorage:WaitForChild("RemoteFunction") -- "RemoteFunction" must be replaced with the name of the remote function in replicated storage

remoteFunction.OnServerInvoke = function(player)
    local doublePlayerSpeed = speedFolder:FindFirstChild(player.Name)
    assert(doublePlayerSpeed ~= nil or doublePlayerSpeed:IsA("BoolValue"), "Player double speed value not found") -- errors if the BoolValue was not found/it isn't a BoolValue.
    if doublePlayerSpeed.Value == true then -- if the speed is doubled
        local character = player.Character or player.CharacterAdded:Wait() -- makes sure "character" variable is not nil
            character.Humanoid.WalkSpeed = 16 -- changes the speed
            doublePlayerSpeed.Value = false -- changes the BoolValue's value
        return true -- returns the condition the BoolValue was found in.
    elseif doublePlayerSpeed.Value == false then -- if the speed is not doubled
        local character = player.Character or player.CharacterAdded:Wait()
        character.Humanoid.Walkspeed = 25
        doublePlayerSpeed.Value = true
        return false
    end
end

-- IF SAME SCRIPT AS EARLIER PLAYERADDED SCRIPT:
local remoteFunction = game.ReplicatedStorage:WaitForChild("RemoteFunction") -- "RemoteFunction" must be replaced with the name of the remote function in replicated storage

remoteFunction.OnServerInvoke = function(player)
    local doublePlayerSpeed = speedFolder:FindFirstChild(player.Name)
    assert(doublePlayerSpeed ~= nil or doublePlayerSpeed:IsA("BoolValue"), "Player double speed value not found") -- errors if the BoolValue was not found/it isn't a BoolValue.
    if doublePlayerSpeed.Value == true then -- if the speed is doubled
        local character = player.Character or player.CharacterAdded:Wait()
            character.Humanoid.WalkSpeed = 16
            doublePlayerSpeed.Value = false
        return true
    elseif doublePlayerSpeed.Value == false then -- if the speed is not doubled
        local character = player.Character or player.CharacterAdded:Wait()
        character.Humanoid.Walkspeed = 25
        doublePlayerSpeed.Value = true
        return false
    end
end

1. Invoke from the local script in text button if the text button was clicked (remote functions return a value/values depending on what you return in the server script/local script):

local remoteFunction = game.ReplicatedStorage:WaitForChild("RemoteFunction") -- "RemoteFunction" must be replaced with the name of the remote function in replicated storage

script.Parent.MouseButton1Click:Connect(function()
    local doubledSpeed = remoteFunction:InvokeServer()

    if doubledSpeed == false then -- checks if the speed is not doubled
        script.Parent.Text = "2X Speed: On"
        script.Parent.BackgroundTransparency = 0
    elseif doubledSpeed == true then -- checks if the speed is doubled
        script.Parent.Text = "2X Speed: Off"
        script.Parent.BackgroundTransparency = .5
    end
end)

1. Add an auto-speed setter script to set the speed if the player dies (SERVER SCRIPT):

-- IF SEPARATE SERVER SCRIPT FROM PLAYERADDED SCRIPT:
local speedFolder = game.ServerStorage:WaitForChild("SpeedFolder")

game.Players.PlayerAdded:Connect(function(player) -- if the player is added
    player.CharacterAdded:Connect(function(character) -- if the character is added for the first time
        character.Humanoid.Died:Connect(function()
            character = player.CharacterAdded:Wait() or player.Character -- when the player respawns, character isn't nil
            local speedDoubled = speedFolder:FindFirstChild(player.Name)
            assert(speedDoubled ~= nil or speedDoubled:IsA("BoolValue"), "Double speed value not found") -- checks if the value is not nil or it isn't a BoolValue
            if speedDoubled.Value == true then -- checks if the speed is doubled
                character.Humanoid.WalkSpeed = 25 -- sets the speed value
            elseif speedDoubled.Value == false then -- checks if it's not doubled
                character.Humanoid.WalkSpeed = 16 -- you might not actually need the elseif statement
            end
        end)
    end)
end)

EDIT: 7. Remove the values if the player is leaving to prevent duplicates/memory being wasted (SERVER SCRIPT ASWELL):

-- IF SEPARATE SERVER SCRIPT FROM FIRST PLAYERADDED SCRIPT:
local speedFolder = game.ServerStorage:WaitForChild("SpeedFolder")

game.Players.PlayerRemoving:Connect(function(player)
    local plrBoolValue = speedFolder:FindFirstChild(player.Name)
    assert(plrBoolValue ~= nil or plrBoolValue:IsA("BoolValue"), "Double speed value doesn't exist (2)") -- errors if the bool value is nil or it isn't a BoolValue
    plrBoolValue:Destroy()
end)

ALL DESCRIPTIONS ARE IN THE SCRIPT/INSTRUCTIONS. Hope this helps!