This has been called a "started logarithm" by some (e.g., John Tukey). (For some examples, Google john tukey "started log".)
It's perfectly fine to use. In fact, you could expect to have to use a nonzero starting value to account for rounding of the dependent variable. For example, rounding the dependent variable to the nearest integer effectively lops off 1/12 from its true variance, suggesting a reasonable start value should be at least 1/12. (That value doesn't do a bad job with these data. Using other values above 1 doesn't really change the picture much; it just raises all the values in the bottom right plot almost uniformly.)
There are deeper reasons to use the logarithm (or started log) to assess variance: for example, the slope of a plot of variance against estimated value on a log-log scale estimates a Box-Cox parameter for stabilizing the variance. Such power-law fits of variance to some related variable are often observed. (This is an empirical statement, not a theoretical one.)
If your purpose is to present the variances, proceed with care. Many audiences (apart from scientific ones) cannot understand a logarithm, much less a started one. Using a start value of 1 at least has the merit of being a little simpler to explain and interpret than some other start value. Something to consider is to plot their roots, which are the standard deviations, of course. It would look something like this:
Regardless, if your purpose is to explore the data, to learn from them, to fit a model, or to evaluate a model, then don't let anything get in the way of finding reasonable graphical representations of your data and data-derived values such as these variances.