This is a perfect example of why you don't hardcode connection strings!
You need to provide a connection string with a full reference to teh instenace of SQL server, which normally will be something like "COMPUTERNAME\SQLINSTANCE NAME".
For example, one of mine is:
DataSource=GRIFF-DESKTOP\SQLExpress;Initial Catalog=...
You can only normally get away with just the instance name on local DB's
You also need to check that the SQL Server instance is configured to allow remote connections as this is often disabled for security reasons.