In this case, i think "Zu Hause" might mean more than "the place you live at", but refers to "home" in a possible meta-sense, since internet cafés are usually used by people on vacation, possibly from another country. Thus "sich zu Hause in angenehme Erinnerung bringen" means something along the lines of "make the people at home positively think about oneself".
As stated by the other answers, this is a pretty rare phrasing, but still legit. I doubt that the meaning is about the guy visiting the internet café thinking of home, but rather telling the people at home about one's well being.
When i was young you normally used post cards for that, but since we are living in a digital age now, posting on facebook, sending an e-mail or using a instant messenger should be the means the author of that notes is thinking about.