How can I make a part after touching break afterwards?

The problem is:

script.Parent.Touched:Connect(function(hit)
    local debounce = true --When the function starts, the debounce is set to true
    if debounce then --So this part is useless, because it's set to true in the prev. line
        debounce = false
        if hit and hit.Parent and hit.Parent:FindFirstChild("Humanoid") then
            local player = game.Players[hit.Parent.Name]
            player.leaderstats.Minutes.Value += 1
        end
        debounce = false
    end
end)

You can do:

local debounce = true --I moved it up, so it won't be to true every time the function starts
script.Parent.Touched:Connect(function(hit)
    if debounce then
        debounce = false
        if hit and hit.Parent and hit.Parent:FindFirstChild("Humanoid") then
            local player = game.Players[hit.Parent.Name]
            player.leaderstats.Minutes.Value += 1
        end
    end
end)

Or you can Destroy the part when it's touched:

script.Parent.Touched:Connect(function(hit)
    if hit and hit.Parent and hit.Parent:FindFirstChild("Humanoid") then
        local player = game.Players[hit.Parent.Name]
        player.leaderstats.Minutes.Value += 1
        script.Parent:Destroy() --This will delete the part permanently
    end
end)

Let me know if it helped

Well, I can suggest you a better method, but it can be only executed once per server. It can be done by using Disconnect. Later, I will also fix your debounce, so it won't error if you try using debounce next time.

Advantages of Disconnect : You can still add code to your script, and make it run (if not connected with Touched event). It just makes a connected function disconnect but won't affect the rest of the code (out of that function).

The simple fix :

local connection -- Creating a nil variable
connection = script.Parent.Touched;Connect(function(hit) -- Setting that variable to the function
    if hit and hit.Parent and hit.Parent:FindFirstChild("Humanoid") then
        local player = game.Players:GetPlayerFromCharacter(hit.Parent) -- It is a better way to get the player
        player.leaderstats.Minutes.Value += 1
        connection:Disconnect() -- Disconnects our functions and makes it never to be used again
    end
end)

Now, here's the script fixing your debounce :

local debounce = true -- MAKE SURE, DEBOUNCE SHOULD BE OUT OF THE FUNCTION THAT YOU ARE APPLYING ON

script.Parent.Touched:Connect(function(hit)
    if debounce then
        debounce = false
        if hit and hit.Parent and hit.Parent:FindFirstChild("Humanoid") then
            local player = game.Players[hit.Parent.Name]
            player.leaderstats.Minutes.Value += 1
        end
    --  debounce = false [This line is useless, as you have already set your debounce to false]
    end
end)

-- Well, the error might happen, when another part touches this part, due to which, Touched event will fire and setting of your debounce to false. The result will be, if player touches it after another non-humanoid part have touched it, then the value won't increase.

Lemme know if it helps!