I recently overheard a discussion that should have gone extinct years ago:
"We try to update our home page twice a month … you know … to keep it fresh"
In my early CMS days I used to hear this type of thing all the time when people were trying to justify the need for a CMS by quantifying how much they edit the site. Even then, I thought the idea that people would be pleased by homepage "freshness" was a quaint little fantasy. In the real world, nobody who matters cares about the fact that a home page has changed. The absurdity of the idea is reflected in this Onion article "Man Cruises By William H. Macy's Website To Check Out The Latest" (hat tip to Deane Barker for mentioning it at the Now What Conference).
There is only one reason to change your home page and that is to make it better. What do I mean by better?
Obviously, you don't want to have any outdated information on the homepage. If you were promoting an event, you want to replace that promotion when the event has passed. You may even want to replace it when it is too late for an attendee to consider going. If you have a new product that you want to push, that product should get top billing. If you have changed the way you describe your organization, that change needs to be reflected in the language on the home page. Being current is keeping up with with your content marketing strategy.
As I have said before, the only reason why someone visits your home page is because he does not know where to go. More intentional visits will come from deep links from search engines and other sites. Use this indecision to your advantage by driving the visitor to your goals. What content is the most compelling? You will never know until you test and that is where analytics and multivariate testing comes in. By the way, never do simple A/B testing. Always use an "epsilon-greedy" approach that allows the more effective option to automatically bubble to the top.
The simple rule is to never change content unless you have something better to show. But just because the rule is simple, it doesn't mean it is easy to practice. To follow the rule, you need to be producing current content that is in line with what your organization is doing; and you need to be hyper-aware of how your content is performing so you can continually improve it. Capriciously changing for the sake of change may make you feel productive and dynamic, but it isn't doing your organization any good. Never changing your home page is not a good thing either. It is a symptom of a stagnant organization and that is not something that you fix by regularly scheduled homepage changes.