I think the main reason why some sayings use the 'fico secco' or 'ficosecco' words to indicate valuelessness is that the fig plant is an endemic and fast growing plant. It is not unlikely (it was more common in the past, though) to find this plant in the countryside, in nobody's land, carrying abundant harvest withering on the plant with nobody caring to harvest them.
The fig tree is diffused in temperate and warm climates, in Italy and in many other regions, yet robust enough to stand freezing winters. It needs little water and prefers calcareous and poor soils. Contrary to most fruit plants in the Mediterranean area, it is not grafted, and does not require any care to bear plenty of fruits. If you want to have your own fig tree, you do not buy seeds or a small plant to grow, just throw a couple of fig fruits in the ground (nearby a plot surrounding wall for example, as it prefers calcareous soils) to have a fruit bearing tree in just a few years.
There might be other, may be less important but supporting, reasons.
One could be that although the dry fig fruits you buy are nice looking (and somewhat expensive), when on the tree they look withered and wrinkled, giving a feeling of somethind passed-away and disposable.
A second might be that the 'c' (k) and 'cc' sequence in the two words give it a rather sharp pronunciation, a pretty distinctive feature in the Italian language.