After the offering has been taken, the congregation sings a song that seems to serve two purposes. Firstly, it expresses thanksgiving for the gifts God has given us. Secondly, it marks the transition from The Service of the Word into The Service of the Sacrament.
At First Lutheran and at many other Lutheran churches there are a few options for the Offertory. The first draws from Psalm 116 asking "What shall I render to the Lord for all His benefits to me?" (the expression of thanksgiving). It also says, "I will take the cup of salvation" (at least a serendipitous allusion to the Lord's Supper).
The next option draws from Psalm 51. This gets at the preparations for the Lord's Supper as it asks God to create clean hearts within us and to renew our spirits.
The next option asks God to "let the vineyards be fruitful." Such a statement asks God to make good on His promises to bless those who have poured out their full tithe (Malachi 3:10). But, like the first option, there is Lord's Supper language as we ask to "be fed with the bread of life" and for God to "grace our table" and "give us a foretaste of the feast to come."
In this way, the offertory serves as an interlude, a transition. We respond to God's Word and promises by offering sacrifices to Him. These are not sacrifices of bulls and sheep and goats that are supposed to atone for sin, but rather sacrifices of a different kind. Sacrifices of thanksgiving that are the first fruits of the abundance God has given us.
Simultaneously, we are preparing to celebrate the Lord's Supper where we recall Christ's sacrifice for us - His body and blood, given and shed, for us and our salvation.
The offertory is like halftime in a way. Halftime is a time to review the first half, and prepare for the second half. You review what went wrong and right in the first half. You make adjustments and go forth with a game plan into the second half.
In worship, the first half is the Service of the Word. The second half is the Service of the Sacrament. The offertory responds to the first half and prepares our hearts for the second half. The offertory says thank you to God's Word of Law and Gospel (what went wrong and what went right), and marches into the second half with full confidence in the victory of Christ that is passed down to us week after week.
Calling it "Halftime" in the hymnal probably wouldn't fly, but I like it.