Loser's Guide to Life
A comment at Twenty Major mentions this behaviour, doing a lot of things at once (messaging, listening to music, talking on the phone while watching youtubes).
The term is recorded at Word Spy, which concludes that it was coined by Microsoft executive Linda Stone as long ago as 1998.
I seem to recall getting shouted at a lot for this while in school. I don't think it would have done to say: “But I am paying some attention, up to a point, much of the time.”
On the other hand, it's not a new thing. In their normal state people are always paying partial attention to a great number of things, taking in potentially useful information. If you're walking down the street you notice traffic, pedestrians, new construction, the weather, etc. What's new is people get to do this at work.