Whether you need a log2FC to be >1 is up to you.
In most cases it is not just about what we feel, but what we can convince others to be relevant. Anyone can feel great that the expression of their gene changed 1.17x from one condition to another, but we still need to convince our peers (and reviewers) that this change is biologically meaningful.
Obviously if your experiment is overexpressing gene A under a super strong promoter, then I'd be concerned about getting a log2FC of +0.31.
I don't think rationalizing log2FC in terms of the promoter strength is what will help with data explanation. A gene can be 10x overexpressed from a weak or from a strong promoter, because the conditions for high transcription were created (say, all activating transcription factors were present in a given condition). What we are measuring is a relative expression change for a large number of genes with changing conditions, and promoter strength doesn't really enter my thinking. Rather, most people think of the effect of transcriptional noise, and try to rationalize what level of fold-change is likely to raise above the transcriptional noise. I don't know if anyone has quantified this properly - chances are that someone has. Most people will intuitively accept that a fold-change of 2+ is real. One can certainly make an argument that a fold change of 1.7 is real as well, but that argument is less likely to fly the closer we get to 1. In my mind that has nothing to do with promoter strength.