Visual Studio not showing server name when adding connection
Go to SQL Server Management Studio to look for the correct Server Name and copy the value. Make all the configurations by yourself at the end just click Test connection button. If the connection is test successfully proceed with your work.
If the database is in your local machine, Just put a . (Dot) in Server Name and select the database.
run services.mscfind " SQL server browser " , it might be disabled , start that service and set it to automatic.