Cohen's d for regression coefficient? I may be confusing two different concepts in this question, but is it possible to calculate a Cohen's $d$ for a regression coefficient? Could its value represent the standardized mean effect of a one-unit change in the predictor upon the response?
 A: Cohen's d does have similarities with the standardised mean effects of an independent variable on the dependent variable in a regression.
Yet, they are completely different. 
Why they are similar
You could make the claim that Cohen's d measures the standardized effect of some treatment between a control group (sample 1) and a treatment group (sample 2). You may read treatment very broadly, the 'treatment' may be a difference in time or 'attended college'. If you were to run a regression with the 'treatment' dummy variable as an independent variable, the coefficient on that dummy represents the average different between sample 1 and sample 2. If you then correct the magnitude of the coefficient with its standard error (like you do when you calculate the t-statistic), you have something very similar (the same?) to Cohen's d.
Why they are completely different
As soon as the predictor variable is not a dummy variable, the similarities stop. Trying to extend "Cohen's d-logic" to another kind of predictor variable would be wrong. Even if the predictor variable is a dummy, the inclusion of other predictor variables may cause large disparity between the two measures (Sidenote: the size of the difference depends on the degree of collinearity between the dummy variable and the predictor). 
Additionally, the Cohen's d is used as a basis for discussion, whereas regression coefficients are usually used for inference and/or prediction:

Calculating Cohen's d provides useful information for discussion (e.g., allows ready comparison with meta-analyses and the size of effects reported in other studies). Where you are reporting about differences between two means, then a standardised mean effect size (such as d) would be an appropriate accompaniment to inferential testing. (From Wikiversity: Cohen's d)

