early in the morning: the opening stretch of the day, running from before first light through sunrise
This word does one job: it fixes an event at the front end of the day. It was an ordinary time-word in everyday Greek, and the Gospel writers lean on it hard, because so much of what they report happens before the day is properly under way.
Its different uses are not different ideas so much as different jobs for one idea. Usually it tells you when something was done, as when Jesus gets up early and goes out to pray. Sometimes it names the morning itself as a block of the day, especially with "the" in front of it or after a preposition, as in "from morning until evening." Once, in Mark's list of the four night watches, it is the label for the last stretch of the night, from roughly three to six.
The edges of the word are soft. Mark 1:35 stacks it with "while it was still night," and John 20:1 with "while it was still dark," yet Mark 16:2 pairs it with the sun already up. So it covers the whole span around first light rather than pointing at a clock reading. Greek had a sharper word for the daybreak moment itself (ὄρθρος), and a noun for the morning as a thing that arrives (πρωΐα); at the far end of Mark's watch list stands the word for the late hours (ὀψέ).
In the Gospels the word carries weight without meaning anything unusual. Jesus prays before dawn, the withered fig tree is spotted on the way into the city, the council convenes at first light, the leaders bring Jesus to the governor's headquarters while it is barely day, and the women reach the tomb at the same early hour. That whole pattern rides on one small adverb.