In scripture this season between the Feast of Firstfruits and the Feast of Weeks remains unnamed. But God commands us in scriptures to "count for yourselves" (Leviticus 23:15-22) days between the Feast of Firstfruits and the Feast of Weeks.
"Counting the Omer" earned it’s name by the offerings we were commanded to bring into the temple, both before and after, on the the Feast of Firstfruits and on the Feast of Weeks. An omer is simply an archaic means of measuring volume. It measures at one tenth of an ephah (another archaic measurement equaling about 5 gallons). So an omer is about one quart or two cups of grain.
The Lord required an offering of 2 tenths of an ephah of grain for both the Feast of First Fruits and for Shavuot (Leviticus 23: 13 & 17). An omer is 1 tenth of an ephah so the offerings for each Holy Day were equivalent to 2 omers. And so this season of the omer is a season bookended by offering a “double portion” or “double tithe” of our best and our first of our harvest to God.