I used to be a big proponent of weekly communion and I think a strong case can be made for it (as well as some bad cases, too). I think it is healthy for the church to practice communion more often than not. Further, some of the original American Evangelical and TR objections to communion were silly and even worse than stupid. Saying such rot like “Having it too often is like Roman Catholicism.” Or better: it just won’t be special any more.
That said, I don’t see anything in Scripture mandating weekly communion. At best, one can make an argument for daily communion, but not one for weekly communion. On the other extreme, I think quarterly communion is a bit rare. One thing that came to mind was communion twice a month. Many Reformed churches are doing it monthly, and that is exponentially better than doing it quarterly, but what if someone is sick or out of town on communion Sunday? Then it will be at least two months between communings. Twice a month could help remedy that.
Another point: Some Scottish churches, baptists like Spurgeon, and others set up real tables as needed so the people could sit when communion was being served. I am not advocating that, but I think we can find Scriptural warrant for it.