Abarim Publications' online Biblical Greek Dictionary
The name Ζευς (Zeus) belongs to the chief deity of the Greeks, who was known to the Romans as Jupiter. The genitive of Zeus is Dios (Διος), which corresponds to an ancient Proto-Indo-European root that expressed brightness of sky and secondarily clarity of vision. From that root come the familiar words dio and deus, meaning god, divine, meaning godly, and diva, meaning deified (feminine), as well as the name of the Vedic deity Dyaus Pitra (Bright-Sky Father).
Several New Testament names derive either from the name Zeus, or from its parent root (see below), and one peculiar adjective: διοπετης (diopetes), which combines διος (dios) and either the verb πετομαι (petomai), meaning to employ wings (to fly and thus see very far, or to cover and thus protect), or the verb πιπτω (pipto), meaning to fall. It denotes something both divine and falling or having fallen, and appears to have been used for sacred objects that apparently had come falling out of the sky. Apparently, the link between the divine and the act of falling was not uncommon in the old world: the familiar name YHWH also combines these two.
Our adjective is used only once in the New Testament, namely in Acts 19:35, which reads: "the city of the Ephesians is guardian of the temple of the great Artemis and of the [or: that being the] διοπετης (diopetes)." Also note the obvious similarities with the qualities of both Jesus (Colossians 1:15, John 3:13 and 6:38), satan (Luke 10:18, Revelation 12:9) and Lucifer (Isaiah 14:12).
There are as many opinions as there are experts on the nagging question whether our English word "day" comes from the same ancient root, which would make sense but is not necessarily true. The same goes for the familiar Greek word θεος (theos), also meaning god. Whether the latter is technically related to deus will probably remain unclear for good, but some say that theos rather relates to the English verb "to do".
That "day" expresses that part of the 24-hour cycle during which most work is "done" may be utter coincidental, directly derivative, or somewhere in between. The relation between these words may be comparable to the false connection between our words "god" and "good", which in modern English look and mean so much alike that an etymological relationship seems inevitable, but which in reality stem from different concepts and wholly different terms.