The "eyeball" you describe does sound somewhat like the "anchor" icon that appears when you click the wheel switch (press down on the wheel) on a "wheelmouse." It looks like a "fuzzy circle," with a "dot" in the center, a triangle pointing up above the dot, and a triangle pointing down below it.
The intended use is - you click the wheel to "place" the anchor, and then the screen scrolls when you move the mouse off the anchor. (Scrolling can be very, very, very fast this way.) You click the wheel again, or hit any key, to "erase" the anchor.
The "anchor" can be placed with some 3-button (or more button) mouse kinds using an assigned button and/or some "combination" of buttons.
If you have a "wheel" on a two-button mouse, a fault in the switch that senses a "wheel click" can turn on the anchor pretty randomly. It's also possible that a keyboard with a "sticky key" can supply the "bucky-bit" to make your computer think you've pressed a wheel when you click one of the other mouse buttons.
If you DON'T have a wheel on your mouse, you need to go to Control Panel and double-click on your Mouse and make sure that PnP has sensed your mouse as a "no-wheel" type. If you see a tab for setting what the wheel does, you've probably got the wrong mouse driver installed.
If you do have a wheel on your mouse, I'd suggest a thorough cleaning of the mouse, although it can be extremely difficult to get crud out of this switch on many of them. You could also try temporarily using a different keyboard to see if you're getting problems from a sticky key there.
A "hanging" switch, either in the mouse or in the keyboard could cause the disabling of keyboard and/or mouse functions, since the machine is waiting for you to "finish the click."
On the other hand, if this is a different kind of eyeball (bloodshot?, crying?, winking?) we'll need another SWAG.