This depends on the network you are in, there might be some services that rely on network names to identify a computer. But as long as you are not in a big network with windows servers I doubt that there is anything that really relies on computer names instead of IP addresses or other, safer ways of identification (eg certificates).
If you have no services in your network that need to access your computer (I mean literally, they want to access your computer remotely), you should not have to worry. This counts for most networks, your home network or public networks (university, coffee shop).
At your workplace you should ask the network administrator but most likely it would not be a problem there either.