Finally my library history and home decorating knowledge can combine! In the decorating world these are called banker's lamps or "Emeralite" lamps, they were a very popular style of lamp at the turn of the century, there is nothing especially library-like to them, and the style is enduringly popular in home decorating. They were a popular, efficient form of task lighting at the time these libraries were likely to be fitted for electric lighting, really nothing more to it than that. They stick around because they're rather handsome little lamps! Green is believed in our society to be a color soothing on the eyes (like green rooms for actors), although I think scientifically the benefits of colors on vision have been shown to be bunk, so it's not reducing any eye-strain. Modern-built libraries generally do not use task lighting any more as it's not energy efficient (who wants to run around turning off 50 lamps before they close the library?). I'm bathed in florescent horror as I work right now. I've seen some modern carrels (those are those cubby things you only see in libraries) that have task lighting, but they're those standard cubicle florescent under-mounted things.

The company that made them has a history page, it's not bad. They came in other colors but I guess green was the most popular. It's a pretty shade of green!

To clarify, the green shades are open at the bottom so that "white" light is thrown onto the table and reading surfaces. Remember that incandescent bulbs throw out light in all directions. The shades prevent bright white lights from also being in readers' peripheral vision, distracting or partially blinding them.

Hopefully I am not too late for this, but I believe the most historical answer lies within the text of Isidore of Seville's Etymologies.

According to Isidore, "Architects use green Carystean marble to panel libraries, because the green refreshes weary eyes."

Sources: Etymologies, VI.xi.2