Is it best practices to display/not display a dropdown menu for the Home button in menu navigation?
(If this isn't just an abstract idea and you have an actual situation to deal with at the moment, maybe you should narrow the question down a bit with that.)
Generally speaking the "home" concept only has a single function that reasonably goes along with it. I don't think I've ever seen a home menu item with sub-items, but any rule in UI design tends to have an implied "unless there's a good reason to do otherwise" on the end of it.
It's not a common convention to display a drop down menu under a Home link, so it's probably better to avoid it. Navigation is best presented as a distinct and obvious list of links, not a hidden area that only a small portion of your audience may discover.
Some would go further and argue that no 'Home button' is needed at all; it is a common convention to link your logo to your homepage, so I'd suggest that you follow that instead and present any drop down menus under clear headers in an obvious navigation area, like eBay, Amazon, Apple, and most of the sites listed in Alexa's top 500 do.
When trying to determine best practices, I always start with the 'best' sites and see how they've solved basic user experience problems. They'll have invested heavily in discovering what works.