The Uncredible Hallq comments on thought experiments. The Stanford Encyclopedia of Philosophy is usually pretty good, but in this case, it falls down on the job. They miss the most interesting examples in actual science, which show the real value of thought experiments in uncovering hidden assumptions and sometimes contradictions in our formulation of physical laws.
We have to dismiss Einstein's "foundational" thought experiments, because they assume — they do not prove — the equivalence principles that you can distinguish neither between uniform motion and stasis (Special Relativity) nor between constant acceleration and a gravitational field (General Relativity). They're not really thought experiments, they merely illustrate Einstein's assumptions.
The only thought experiment with any value discussed in the article is Galileo's (supposed) proof by contradiction that all objects fall at the same acceleration. However, Galileo's thought experiment is merely suggestive; proves only that Aristotelian gravitation is incomplete and not rigorous about what constitutes an "object". It's notable that (at least according to legend) Galileo actually performed physical experiments to test his ideas.
Missing from the article, though, are three very important thought experiments: the Brownian Ratchet, Einstein's Box and the EPR Argument. In all of these cases, the original argument proposes a contradiction in the formulated laws of physics, the Brownian Ratchet in thermodynamics, and Einstein's Box and the ERP argument in quantum mechanics. In the first two cases, the apparent contradiction was resolved by discovering non-obvious lack of rigor in the thought experiment, and the paradox resolved. In the last case, Einstein, Podolsky Rosen actually demonstrated that quantum mechanics does not permit local hidden variables*, a finding which has been explicitly stated and proven by physical experiment.
*or other equally prosaic assumptions about classical reality.
In all cases, we see thought experiments in science to do one of three things: illustrate assumptions, examine real or apparent contradictions in a theory, or discover non-obvious implications of a theory. A thought experiment in science can at best act as disproof, never as a positive proof.
It's hard to understand Norton's empirical view (as described in the article) of scientific thought experiments, that "any thought experiment is really a (possibly disguised) argument; it starts with premisses grounded in experience and follows deductive or inductive rules of inference in arriving at its conclusion." It's ambiguous what is meant by "premisses grounded in experience"; in the thought experiments cited above, the premisses are grounded in the theory (thermodynamic or quantum mechanical), and they teach us about what the theory says (or we think it says), not whether the theory is actually true.
On the other hand, it seems easy to reject Brown's view (as described) that, "in a few special cases we do go well beyond the old data to acquire a priori knowledge of nature, ... [because] Galileo showed that all bodies fall at the same speed with a brilliant thought experiment." Galileo showed (proved) no such thing; he showed only that Aristotle is ambiguous about what constitutes an "object". It's not actually the case, but it might have been the case that if we tie a string between a heavy object and a light object, they will together fall faster than either alone. If we can tell the difference between attached, compound objects and detached, independent objects, it's not logically impossible that the universe can tell the difference as well. Weird, perhaps, but not impossible. (And if Aristotle had said density instead of weight, he would have been correct, at least in an atmosphere. If we tie two objects of unequal density together, they really will fall at an acceleration proportional to their average density.)
Thought experiments in philosophy do exactly the same thing as they do in science: They either illustrate how we think, or they expose apparent or real inconsistencies in how we think.
The problem in philosophy, though, is that it's monumentally uninteresting to show yet again that human beings do not ordinarily think with precise logic. We take all kinds of shortcuts, some of them quite effective as heuristics, many of them simply stupid. Socrates built his entire reputation on nothing more than proving that people don't think precisely.
We prove nothing interesting by showing that people can think one way about this problem and another way about that problem even though the two problems share some important feature in common. The Famous Violinist Problem just shows that differences in our thinking about violinists and abortions (if one does think differently about these things) are due to the differences in the two cases, not their similarities.
Of course, there are thought experiments that are just bad, that mislead rather than illuminate, such as Searle's Chinese Room. The problem with the Chinese Room is that Searle is ambiguous and imprecise about what he means by "understanding". By specifically putting a non-Chinese-literate person in the room, he's inviting the reader to look for the understanding in the person; since the person ex hypothesi doesn't understand Chinese, he concludes that no understanding is present. At best the Chinese Room proves only that we don't understand what it means to understand something, at least not rigorously.