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".