Now concerning the collection for the saints: as I directed the churches of Galatia, so you also are to do. On the first day of every week, each of you is to put something aside and store it up, as he may prosper, so that there will be no collecting when I come. And when I arrive, I will send those whom you accredit by letter to carry your gift to Jerusalem. If it seems advisable that I should go also, they will accompany me. – 1 Corinthians 16:1-4
Paul’s collection is for disaster relief, not for Paul’s ministry. And so, for accountability purposes, he wants that money to touch his hands as little as possible. When he comes, he wants the money to already be allocated in each person’s budget, if not already collected, so that there isn’t even the appearance of him speaking for a fee. When the money gets sent to Jerusalem, where the disaster is supposed to be, he wants it to travel in somebody else’s pocket.
It’s completely incidental that the collection day is Sunday. Apparently that’s just the most obvious time for setting aside cash. Either that’s because people get paid on the last day of the week, or because they meet for worship on the first day. But it’s in the Bible, so now it’s self-reinforcing, and we all worship (and collect offerings) on Sunday, regardless of when we get paid.