I would suggest using a microarray analysis software to define the differences statistically. There are many such packages, so try checking at your institution to see what commercial software is available. Alternatively, there are many free alternatives (I am partial to the limma package from the bioconductor project). Using a statistical test reduces both your false positive and false negative rates, assuming that you have a few replicates for each tissue.

I don't have access to my data and the references therein at the moment. Much of the expression data I have collected (from published sources), both for mRNAs and miRNAs, uses metrics like FDR to assess significant differential expression. If FDR is low, then a 10% change in expression can be significant, and trustworthy for building an explanation and further hypotheses. Much of the data I have use FDR or similar approaches and show changes in expression from control of +/- 10-15% and higher. So, 15% can be fine if the change has low FDR or is significant or low variance.

The other two answers here are good in that they talk about ways to get statistical significance, but I'm not sure they really address the "tissue-specificity" question you're asking. First off, defining "specificity" based on a survey of four tissues seems pretty tenuous to me (though it sounds like you're aware of this). And you're right, anything thresholds you choose will be rather arbitrary, so ideally they are chosen with an eye to how the results will be used (tolerance to false positives and false negatives).

If your primary interest is in using criteria that are defensible relative to a published study, you could reference our previous paper here. But wow, talk about rudimentary criteria -- almost painful to look at now.