What is the scientific method for finding a theoretical explanation for experimental data? Say you observe some effect that you did not anticipate in one of your experiments. You have checked the experimental setup and you're sure it's not just a systemic error but an actual measurement of something. You are also able to repeat the measurement, and the signal behaves the same way.
Of course, you have some basic theories that describe your experimental setup, but since you are working on something that has not been observed before, there is no literature available, and there are a multitude of physical effects that could technically be involved. (It is possible that some other experiment could give more insight, but it would require a completely new lab setup and you don't know any scientific groups that have such a setup.)
In this situation, what is the method that should be used to proceed and, if possible, to get a clear result?
My personal gut reaction is to try to fit the data and to get some information out of that, but this only works well if the fit function happens to be correct. I can fit anything with polynomials, but they do not tell me anything about the physical cause of the observation. And even if the fit function is correct (e.g. if the observation is an exponential decay, and you are able to fit that), it is not always possible to identify the origin correctly.
Is it maybe the best idea to give up trying to explain the observation, and simply mention in a paper that you observed it? Or is there some typical method that is applied in these circumstances that I cannot think of?
Of course the ideal case is that one is able to derive a nice theoretical description, which could even be derived before conducting the experiment, and the experiment is then simply evidence that the theory is correct. That's the ideal case, but I feel like that is not what happens in many cases. Both Newton and Einstein needed experimental observations before they could develop their theories, just to name two famous examples. And the whole of quantum mechanics is based on people trying to explain weird observations that do not fit into existing theoretical models. What I would like to know is the type of process that is involved in getting from "that's weird" to "this is the answer".
 A: 
My personal gut reaction is to try to fit the data and to get some information out of that, but this only works well if the fit function happens to be correct.

Whenever you fit a meaningful curve to your data other than a freeform curve, there will be a specific (physically or technically motivated) model behind this curve, which you assume to be correct, when you apply the fit.
It usually does not make sense to fit data of unknown physics with "some" functional form, and to then claim, "we have observed the measured data to follow this function". The reason is that there may be other functions that fit the data equally well, or there may be other functions that do not fit the data quite as good, but are physically much better motivated, i.e., much closer to the true origin of the data.

Is it maybe the best idea to give up trying to explain the observation, and simply mention in a paper that you observed it?

This is always an option at the very end of the process of trying to find out what is going on, when you were not successful to nail down the origin of the effect, but you consider it worth mentioning.

what is the method that should be used to proceed and, if possible, to get a clear result?

The first steps in an experiment with an unknown effect could be to


*

*try to find some "handle" on the unknown effect: Which parameters can I change in the experiment? Which parameters does it depend on, which does it not depend on? Can I perhaps discriminate in this way, if the effect originates from a technical problem in the setup or from the actual physics of the system under study?

*reduce complexity: Can I simplify the setup to exclude some possible spurious influences on the experiment? Did I do everything to make sure that my measurement apparatus has no flaws and works exactly as expected? Can I characterize single parts of my measurement setup independently? Can I simplify the complexity of the measured system (i.e. by modifying the device design or similar)?

*think about fundamental principles: Is the effect in contradiction with fundamental principles? Does it violate certain symmetries? Does this give a hint about the origin of the effect?

*check reproducibility: Do I systematically find the effect in similar experiments, e.g., with different devices? Results that you found once in some sample that you could never reproduce in another may not be of the greatest interest.

