Friedman, Hastie, and Tibshirani (2010), citing The Elements of Statistical Learning, write,
We often use the “one-standard-error” rule when selecting the best model; this acknowledges the fact that the risk curves are estimated with error, so errs on the side of parsimony.
The reason for using one standard error, as opposed to any other amount, seems to be because it's, well... standard. Krstajic, et al (2014) write (bold emphasis mine):
Breiman et al.  have found in the case of selecting optimal
tree size for classification tree models that the tree size with minimal cross-validation error generates a model which generally overfits. Therefore, in Section 3.4.3 of their book Breiman et al.  define the one standard error rule (1 SE rule) for choosing an optimal tree size, and they implement it throughout the book. In order to calculate the standard error for single V-fold cross- validation, accuracy needs to be calculated for each fold, and the standard error is calculated from V accuracies from each fold. Hastie et al.  define the 1 SE rule as selecting the most parsimonious model whose error is no more than one standard error above the error of the best model, and they suggest in several places using the 1 SE rule for general cross-validation use. The main point of the 1 SE rule, with which we agree, is to choose the simplest model whose accuracy is comparable with the best model.
The suggestion is that the choice of one standard error is entirely heuristic, based on the sense that one standard error typically is not large relative to the range of $\lambda$ values.