How to best manage Protoss Observers in StarCraft 2?

If you know the general area your observer is in but are having trouble spotting it quickly, try holding the Alt key. This will make its health and shield bars visible and might help you pick it out (unless it's hovering directly over a large group of ground units, or something).

If the observer is just wondering around, it should be easy to pick it up in the minimap - it will be the only friendly dot in the middle of nowhere, not to mention it will give you vision of everything around.

If, however, it was forgotten next to other friendly forces / buildings, I see no easy way to locate it unless you remember where it was or you notice an unusual field of vision (too large / can see up cliffs). If you only remember the general area where it used to be, you can go there and drag the mouse to select an area, in case it's hiding over a building or something - this will show it in the selected unit list.

Finally - though this is something not really applicable when you're desperately looking for an observer - is that you can bring another observer to the area and then double-click it, that will select all the observers around including the missing one.

So I guess my best tip for you is to not lose them :) keep an emergency observer in a hot-key, it may save you later.

First of all - make a habit to bind all your obs to one button. You can go to specific one by selecting and click on portrait (an option in 'game' settings 'smart camera pan').
If you definitely know that there is observer on current screen - box all units and find obs in group by icon.