From the GoF book:
A flyweight is a shared object that can be used in multiple contexts simultaneously. The flyweight acts as an independent object in each context— it’s indistinguishable from an instance of the object that’s not shared. Flyweights cannot make assumptions about the context in which they operate. The key concept here is the distinction between intrinsic and extrinsic state. Intrinsic state is stored in the flyweight; it consists of information that’s independent of the flyweight’s context, thereby making it sharable. Extrinsic state depends on and varies with the flyweight’s context and therefore can’t be shared.
- Gamma, Erich; Helm, Richard; Johnson, Ralph; Vlissides, John. Design Patterns (Addison-Wesley Professional Computing Series)
A flyweight in a flywheel is used to store mechanical energy to be shared later (much like an electrical inductor or an accumulator).
Additionally, flyweight is a light-weight boxing category.
Considering the definition given by the GoF, I surmise the term was originally chosen as a metaphor for a flywheel flyweight.
However, since flyweight also means lightweight boxing category, they probably called it the "flyweight" pattern instead of the "flywheel"/"inductor"/"accumulator" pattern because flyweight means both a shared object AND a lightweight object.
One can't help but laugh at the irony that mechanical flyweights are actually very heavy, in contradiction to the fact that the term is used for lightweight boxing.
The GoF also states that
The concept of flyweight objects was first described and explored as a design technique in Interviews 3.0 [CL90].
and that paper you can find here. The authors don't explain why they named it that, but it's clear from the paper that they are defining a lightweight, sharable object.