Gilda's Lost Axe is a quest given by Gilda in Mean Street by talking to her before asking about the gear needed to activate the Ostown Projector Screens.
She will tell Mickey that she was climbing Mickeyjunk Mountain when Blotlings appeared, and she had to retreat (leaving her axe on the Mountain in the process). Her axe can be found in the Collector's Hut on the Mt. Osmore Slopes.