I sometimes speak about statistical results to a popular audience, and the term "significant" can (understandably) be misunderstood. I sometimes want to say something like "the likelihood of seeing these results under the null hypothesis is small enough that there probably is something going on here, however the effect size is small enough that it's probably not worth worrying too much about."
Among technical audiences, I believe this could be summarized as "small but significant". Is there a short way I can communicate this to a non-technical audience?
If the audience knows what "statistical significant" and $p le 0.05$ means, there is not much that can go wrong. But otherwise, I really like the fantastic suggestion by Jordan Ellenberg as an alternative to "statistically significant" in general:
[…] "statistically noticeable” or statistically detectable" instead of “statistically significant”! That would be truer to the meaning of the method […] – Jordan Ellenberg in his book "How Not to Be Wrong: The Power of Mathematical Thinking"
Edit based on the short discussion in the comments: Note this answer does not specifically address the "small effect" situation but rather proposes an apt word for statistical significance in general where you don't have to fear that people take "statistically significant" as "relevant". In this way, you can distinguish in a clean and understandable way the two topics "hypothesis testing" and "effect size".
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