I know this is an old question now, but coming from an engineering / policy analysis background I think about these things a bit. So, I think changing an independent variable by one standard deviation is one method to both display the Magnitude of the change and the Likelihood of the change.
For example, people commonly perform sensitivity analyses to determine how some model reacts to changes in a given independent variable. Often, the changes show some degree of sensitivity (e.g., a 10% change in X leads to a 10% change in Y) but such analyses do not provide any indication of how likely a 10% change in X is. You can see how this would become an issue:
Imagine a 10% change in A changes Y by 10% and a 10% change in B also changes Y by 10%. One would think the sensitivity of the model to both parameters is the same (and maybe technically it is) but this ignores the real world. Maybe A is the price of gasoline (which changes frequently and sometimes greatly) and B is the time it takes the earth to rotate around the sun (not going to change by 10%...hopefully). Once we know a 10% change in A is more likely, and thus the model results are more likely to change due to the result of A, we would be more likely to try and improve the model by reducing the uncertainty associated with A.