The eye of God
means the same thing as hand of God
(Ezra 7:6, Ezra 7:9) and refers to divine providence (Deuteronomy 11:12; Psalm 33:18).1 God was busy watching over the Jews; he was working to make sure that they were looked after and that his plans/promises would not be stopped.
5 But the eye of their God was on the elders of the Jews, and they did not stop them until the report should reach Darius and then an answer be returned by letter concerning it.