The name change does not yet show up on the Windows Store listing for Microsoft Health, but the new Microsoft Band name is shown after the latest app update is installed. The name change is visible for the Android version on the Google Play Store page. Aside from the rebranding and unnamed "bug fixes," the app itself does not have any new features, and it still links to the Microsoft Band fitness wearable to keep track of its owner's activities. So far, the iOS version of Microsoft Health has not been renamed.
In addition, we have received reports that the Microsoft Band 2 has received a firmware update. There's no word on what it contains.
This move comes after rumors hit the Internet earlier this week that Microsoft may not launch a new Microsoft Band 3 device this year, and could possibly end making Band hardware altogether. Those same unconfirmed reports claim Microsoft has canceled plans for a version of the Band that runs on Windows 10, rather than custom firmware.
Thanks to everyone for the tips!