For some reason, names drawn from Japanese mythology tend to get used in prosperous times.
Literal
Somehow, economic-condition [gen-の] good time [when-には] Japan [gen-の] myth [to-に] drawn name [subj-が] is-attached [polite-ています].
ちなんだ is the past attributive form of ちなむ 'to be related / linked to,' typically in ~にちなむ 'after X / in reference to X.' Here 日本の神話にちなんだ 'drawn from Japanese mythology' acts as a relative clause on 名前. 景気 specifically refers to economic conditions / economic climate — 景気の良いとき 'times of good economic conditions.' The passive 付けられる 'be attached' with ~ている marks the habitual resulting state.