Store the connection string in your settings file, or in a datafile in the User Data folder (see here for how to use that: Where should I store my data?
To use the settings file, just open your project in the Solution Explorer, open the "Properties" branch, and double click on "Settings.Settings"
Give a new string a name "DatabaseConnection", set it as "string", "user", and set a sensible default (the connection string to your SQL instance)
and save and close the page.
Then when you want to use it:
string conStr = Properties.Settings.Default.DatabaseConnection;
To save a new value:
Properties.Settings.Default.DatabaseConnection = conStr;