A statistical model uses variables to predict the value of another variable (total deaths resulting from the hurricane). The "deviance" is a measure of how much of total deaths is not predicted. So the goal is to get a small deviance using a small number of predictors.
Here are the deviance and number of predictors in two of their models:
136.1 3 female name, storm damage, barometric pressure
121.8 5 "" plus interactions (products) of female and damage, female and pressure
Here are the deviance and number of predictors from two alternative models that I fit:
97.5 3 female name, logarithm of storm damage, barometric pressure
95.3 5 "" plus interactions of female and log damage, female and pressure
The models using the logarithm rather than the original variable had much lower deviance. Adding the two interactions to the model with the logarithm reduced the deviance by 2.2, but the usual standard is that adding two predictors has to reduce the deviance by at least 6 to qualify as evidence that there's anything there (ie a reduction of less than 6 is not "statistically significant"). So the best model has a deviance of 97.5 and three predictors. In that model, the estimated effect of the "femaleness" of the name (which they treat as a matter of degree) is .024, with a standard error of .036, which is not statistically significant, or close to statistically significant.
So the flaw was that they controlled for the dollar value of damage when they should have controlled for the logarithm of damage. With the right control, there is no evidence that the gender of the name makes any difference.
Notes: 1. the paper and data, published in the Proceedings of the National Academy of Sciences
2. Jeremy Freese has a number of interesting comments on the study.