It seems that LAN Messenger doesn't have an option that you can use to save the message history. The only way to do this, is to select all the messages from a session by pressing the CTRL + A key combination (in order to select all the text in the window) and then use the CTRL + C key combination to copy the content. After you did this, you can open Notepad and use the CTRL + V key combination to paste your messages in a new file in order to save them as TXT or HTML. For more information about the program and also suggestions for the developer, visit the official website.