Is there any standard measure of economic significance? One measure is to look at standard deviations. If a one standard deviation increase in X leads to a more than a 0.5 (or 1, or ⅓, or whatever) standard deviation increase in Y, then we say that X has a economically significant effect on Y.

Is this standard? And if not, what other measures are there to formalize the notion of "economic significance"?
Imagine the basic linear regression setup:

$Y_i = \alpha + X_i \beta + \epsilon, \epsilon \sim N(0,\sigma^2)$

Holding everything else fixed, increasing $\sigma$ will increase the standard deviation of  $Y$ and therefore, for a fixed relationship between $X$ and $Y$, eventually move the relationship between them to economic insignificance by the standard you present. But the relationship might still be quite economically important. It might be better to instead ask about the standard deviation effects on $\sigma_\hat{Y}$, the standard deviation of the fitted values of $Y$, to ask if the effect is large relative to the total variation that can be explained by the model.   

In accounting, they use a concept of immateriality which seems quite similar to economic insignificance. That might be a helpful definition for some problems.
This quote from Signifying Nothing: Reply to Hoover and Siegler by Deirdre N. McCloskey and Stephen T. Ziliak may also be helpful:


  The sheer probability statement about one or two standard errors is
  useless, unless you have judged by what scale a number is large or
  small for the scientific or policy or personal purpose you have in
  mind. This applies to the so-called "precision" or "accuracy" of the
  estimate, too, beloved of Hoover and Siegler — the number we calculate
  as though that very convenient sampling theory did in fact apply.