>>>>>It was within literally 10 minutes when the customer sent me email saying that they check everything, as far as firewall or AV and were still getting error. And then I started the SQL Server Browser service and asked him to try again. And he said "everything works now." It is, of course, possible that something else may have occurred at the same time. But I don't know.
>>>>
>>>>When the server was restarted, was it still running? Or, when that PC was restarted, was is still running?
>>>
>>>It was the Service (not server) that restarted. And it was not running. So I simply went to SQL Configuration and Started this Browser service. PC was running and was not shutdown at any time.
>>
>>In case it helps you or anyone else in the future, here are my notes on enabling remote connections:
>>
>>How To Enable Remote Connections SQL Server 2012 Express
>>
>>Open the SQL Server Configuration Manager
>>Expand SQL Server Network Configuration – Protocols for SQLEXPRESS
>>Enable TCP/IP
>>Right click TCP/IP and open Properties
>>Check the IP2 settings reflect the servers IP address and subnet
>>Check IPAll and ensure that Dynamic Ports is blank and make sure that the TCP port is set to 1433
>>Try connecting to the server from a remote machine using SQL Management Studio or your favourite tool, you should be good!
>>
>>If not, expand SQL Server Services from the SQL Server Configuration Manager and change the SQL Browser Service to Automatic and start it, then try again.
>>
>>Also, it’s worth noting that if you’ve already had a fiddle you may need to double check that you have still got Allow remote connections to this server enabled on the instance options. Here’s how to check:
>>
>>Open Microsoft SQL Server Management Studio and connect to the desired instance.
>>Right click on the instance node and click Properties
>>Click on Connections and check that Allow remote connections to this server is checked and you haven’t limited your maximum number of connections over drastically
>>Warning: When stopping and starting SQL services it is worth making sure that the databases are not in use so as not to effect any live systems.
>>
>>To make it work, SQL Browser on the remote server may need to be enabled, UDP port 1434 should be put into exception (on the machine running SQL Server) if firewall present.
>>Also add firewall rule on TCP 1433
>
>Thank you very much.

Dmitry, when you try to connect to a SQL Server you do not need the SQL Browser service if you use the port number when connecting, for example try in SQL Management Studio to add the port as part of the instance you are trying to connect, for example "TheComputer\TheNamedInstance,1433". Now you can not only stop SQL Browser service but you can also close the UDP port 1434 in your firewall as is no longer required. I would also change the port to a different number as port 1433 is known and script kiddies might want to try it, although it might be an overkill this days, but to do it you just follow the same steps but instead of 1433 use a random unused port