Why is the Ethernet network interface name "Network 3" in Windows?

The interface is named "Ethernet"; you can rename it using F2. (Internally, Windows associates configuration to interfaces by their GUID, not name, so you can use any descriptive name you want.)

"Network 3" is how Windows calls the network you're connected to. As part of Windows Firewall, different networks on the same interface are automatically recognized and can be associated with different firewall profiles (home/work/public in Windows 7, private/public in Windows 8).

In other words, this is the third network Windows has been connected to.

To rename the current network or to manage previously recognized ones, go to the Network & Sharing Center, and click the icon next to "Network 3". (Tested on Windows 7, may have changed on Win8.)

Disconnecting and reconnecting the Ethernet interface can cause the interface to be renumbered when Windows thinks the network has changed. This often happens when a laptop moves between networks. This can also happen when the interface changes.

FYI, the name of the interface is actually the top label (Ethernet), the second level is the name of the network (Network 3) and the third label is the type of adapter. On a Corporate network, the network will be AD network name (such as Corp.Bigcompany.com). On a home network, using a workgroup rather than AD, it is just the name Network and a numeric id that increases each time that Windows thinks the network has changed.