In the UUA Rules comes this tidbit:
For purposes of determining compliance with Section C-3.5 of the Bylaws, a member congregation shall be deemed to have conducted "regular religious services" if it has held at least 10 services during the fiscal year.
The fiscal year is July 1 to June 30, but that's not the important part. Section C-3.5 concerns certification, and with it General Assembly voting rights. Being uncertified three times opens your church up to being categorized as inactive, though I suspect the fact of inactivity precedes the administration of it.
So why ten services? Why not twelve, or two?
I think this rule is noteworthy for two reasons. First, it sets a minimal objective standard by which activity can be measured, and so distinguish running congregations from summer chapels -- which persisted until the 1980s -- or from encouraging an entity that uses the name and cachet of "church" to operate some other kind of organization, however useful or well-intended.
And this standard protects congregations from the (unlikely) charge of not meeting regularly enough for worship. I'm referring back to the associational test the IRS has for churches. I don't think this is a coincidence, particularly since one of the questions a church-applicant (and recall a church doesn't need to apply, but may to have a verifiable determination from the IRS) answers (PDF) is
Do you have regularly scheduled religious services? If "Yes," describe the nature of the services and provide representative copies of relevant literature such as church bulletins.
This may not be the main reason, but the floor of ten services does seem to offer cover, and the UUA seems to have good counsel with respect to the Almighty IRS.
Second, I hear an echo to circuit riding, as a former circuit rider myself. Thinking back a few decades (or generations), church could expect to get a minister to lead worship at least once a month. But sometimes harsh weather made summer worship unbearable in the South, and winter worship impossible in the North. And while air conditioning has (largely) solved the first problem, icy roads and the cost of heating oil does little for the second. I'm sure I've seen instances of New England congregations close in January, or move to an alternative facility.
Worship once a month, less the hardest months, sounds like ten services a year to me. Other theories?