While I agree that 'significant' has a misleading connotation, 'discernible' is also misleading. 'Statistically significant' just isn't any everyday concept, and trying to phrase it as one will encourage people to make mistakes. It's a complex concept: if we tried this experiment under a certain null hypothesis, then it'd be at least this improbable to see a result at least this extreme. The most I'd be willing to cut it down, after so much confusion in its actual use, is "subjunctively improbable", with the null hypothesis and the threshold left implicit. "Eating fat had no subjunctively improbable effect on weight gain." This sounds technical and fiddly, which I think is a feature: if you don't like it, don't base your reporting on a technical, fiddly concept.

"Eating fat had no discernible effect on weight gain" sounds like getting evidence against such an effect, but it's compatible with getting evidence in favor, that's just not as strong as some threshold. That evidence could be useful in a meta-analysis, or for a decision when waiting for more information isn't practical or economical, or if the potential gain from trying the nonsignificant treatment is high and the potential loss low. (I've seen "no significant X" abused this way. Nobody should try X -- it's unscientific!)