I ran Eclipse 3.5 (Galileo), and apparently SWT in that version does something wrong in communicating with GTK. The end result is that buttons don’t react to mouse clicks anymore. Rather annoying. Luckily, there is a solution available. Alternatively, you can use the latest Eclipse 3.6 (Helios) milestone.
But that wasn’t the end of it. Eclipse would now perform extremely slowly on a variety of tasks. It turns out that this is caused by Eclipse now running on the GCJ Virtual Machine. I simply uninstalled everything with “gcj” in its name using Synaptic and all was well again.