A back-of-the-envelope calculation by Russell Seitz indicates that the electrons constituting the flow of information on the internet weigh about 50 grams, or less than two ounces. Of course, this doesn't factor in the weight of the computers and physical networks themselves. From the article:
Forbes publisher and blogger Rich Karlgaard recently lamented his $1,200 monthly home utility bill. That's a lot of PG &E , but the yearly power bill for the global internet is just $3 per capita- a bargain even by third world standards. Yet looking at my ISP bill I'm not too happy - a dollar a day seems a trifle high when you reckon the weight of the penny's worth of electricity my computation consumes. My daily fix of electrons in motion costs me about half a billion dollars a pound.