Every algae blenny I ever fed ate a variety of different foods. Algae of course is the main source of nutrition for them and there should always be supplimental algae food like Formula Two as mentioned in my previous post. Algae strips also make a good supplimental algae diet for when natural algae growth is low.
In my experience, if a lawnmower blenny dies of starvation, there are other causes like stress from being in a new environment and refusing to eat and internal parasites robbing the fish of it's nutrients...and of course if the care giver lags on feeding enough food, the fish can starve.