I dont know how much data a router logs, but they might log the mac of all connected devices. so if they see that a specific ip was used in an attack, they simply go to the wifi hot spot and take their router and look up the logs.
Which proves only that the computer that made that transaction was using that MAC address at that time.
even then, going back with a mac address to find the owner would be very difficult. so just buy a cheap laptop at a pawn shop with cash from change and you should be safe.
Or use a random MAC address.