When making a thick simple syrup (1 part water, 1 part white sugar), how can I know when I've cooked the sugar for long enough? Is it possible to overcook this?
You aren't actually trying to cook anything. When you heat a solution, it makes dissolving a solid in to that solution much easier. So you can dissolve more sugar in to hot water than cold water. With a 1:1 ratio, you wouldn't be able to get all the sugar in to solution with cold water. So, you heat the water to allow more sugar to become part of the solution. So the answer is, as soon as there is no visible sugar in the water, you are done.
If you were to cook it for longer, you would reduce the syrup, increasing the concentration of sugar in the solution. You'd have to reduce it an awful lot before you burnt this. However, if you reduce it too much, then the liquid will become solid when it is cooled. Both of those would probably qualify as "over cooked".
Here's a nice science description of sugar solubility. Interestingly, they say that sugar has such high solubility in water that you can get 1800g in to 1L. Using this volume to weight conversion site, that's approximately 7.5 cups of sugar in 1L of water. 1L = 4.2 cups, so you can get ~1.8 Cups of sugar in to 1 cup of water (under ideal circumstances). So, if you are actually interested in cooking your water off to get a more sugary syrup, you can't reduce the water to less than 1 / 1.8 = or 55% of the original volume. However, if you do want a ratio higher that 1:1, then I would suggest introducing the appropriate amount of sugar in the first place and stopping when the sugar is dissolved rather than using less sugar and cooking the liquid down, as the first approach is much more precise.
(for purposes of this answer, I have ignored the possibility of super saturating the syrup, as that would be essentially useless for cooking....although it would be funny)
The answer provided by Yossarian is largely accurate, but it should be emended to point out that cooking is in fact taking place.
Yes, the increased molecular activity resulting from the heat does facilitate the dissolution of the sugar molecules (this is why your sugar doesn't dissolve in your iced tea but does in your hot tea). But most simple syrups are cooked for at lease some time beyond the point where the sugar is simply dissolved. This is especially true of flavored simples where you'd have to steep any herbs or the like for a lot longer than it takes to get the sugar dissolved (I make ginger and mint syrups often, and the ginger in particular requires a good amount of time in the solution before I strain it out).
The other thing to note is that if you continue to cook it for a lengthy period of time, it will become caramel. No, this will not get you traditional caramel, as that requires butter and cream, but it will still darken and go through the soft ball, hard ball, and hard crack stages as it cooks.
Once solution is a candy thermometer, but you really only need that precision if you're trying to get the mixture to one of the above-mentioned stages. A better solution to test your syrup is as follows:
Place a bowl or plate in the freezer while you cook the syrup. When the syrup gets to the point you think you're looking for, remove it from the heat, take the cold dish out of the freezer, spoon a bit of the syrup onto the plate and return it to the freezer for a couple minutes. It will cool rapidly enough that when you pull it back out it will give you a good indication of what your final product will be like. If it's too thin, return the pot to the heat. Too thick? You have to start over.