D) anything else
The null is "accepted," that there probably is no effect. What does "non-rejection" mean?
It can mean two things. First, it could mean you have no knowledge regarding the effect. If you took R.A. Fisher's understanding of a p-value, then the absence of rejection is the same as no added knowledge.
If you approached this from a Pearson and Neyman decision-theoretic perspective, then it means that you should behave as if there is no relationship. From a Pearson and Neyman inferential perspective, however, you simply have no findings.
The p-value gets knocked around quite a bit because people abuse it and ignore effect size.
What it is telling you is that despite the possibly random appearance of a large effect size, it still cannot be shown that it is different from zero.
Imagine I am a rookie major league baseball player and I am batting 800 with five at-bats. My effect size is HUGE, but I cannot falsify that my real batting average is 400 $(p<.058).$
The non-rejection of the null with a large observed effect could mean many things.
First, it could mean there simply is no effect. If there is literature regarding this that there is an effect, then further investigation is warranted. The result could be a false negative. If there is no literature, it could mean that everyone is getting non-significant findings and so no editor is publishing the findings. So it could be that there is no effect.
Second, there could be poor hypothesis or experimental design. It could be that there is an effect but either the hypothesis or the experiment was poorly constructed. For example, in a famous case of Yule's paradox, UC Berkley found that its admissions were gender-biased against women. In fact, the effect was strongly supported though the effect size wasn't necessarily that large. A deeper investigation determined that admissions were gender-biased against men, though again the effect size was small.
The poor experimental design resulted in a conclusion that was the opposite of the facts.
It could be that the experimental design was too simple. For example, there may be some hidden bias among who is an ex-alcoholic OR there could be a hidden bias among who said "yes" to participate. It is possible that gender matters. It is possible that a whole range of things actually matters, but that the division points into the three groups are a poor division.
It could mean that ANOVA was the wrong test. For reaction times, there is a reasonable argument to be made that assumptions are violated.
The difficulty is that Frequentist methods of statistical analysis assert that the null is 100% perfectly true. If you reject the null, then to some degree of confidence, you can assert it is not a true statement. However, if the null is not rejected, that does not imply that it is true.
Frequentist methods of statistical analysis are a probabilistic form of modus tollens. If A then B, and NOT B, therefore NOT A. However, sentential logic does not imply "If A then B, and B, therefore A." In the case of B is true, then A can be true OR false. While it is true that if A is true then B must be true, we know B is true. If A is false, B can still be true. That is why Fisher held that failure to reject the null was the same as having no knowledge about the topic.
Logically, non-rejection of the null implies that either A is true OR A is false. The effect size is irrelevant to that discussion. Since it is a tautology that A is true or false, what you do should not depend on the results of your experiment.
If you believe that there should be an effect, then you could, for your own personal utility, expand your investigation. If you do not believe there should be an effect, you could, for your own personal utility, not continue to investigate. However, the results should not factor into your decision. It is as though you had never decided to look in the first place.