To reiterate what Amit and Subterfuge said above, your connection string should be stored in a configuration file instead of in the code. Those methods won't solve the face that you would still need to modify the connection string on every machine though. Instead, I recommend you create a convention for local development and use configuration transforms between development/staging/production environments.
Local Development Convention
It will be easiest to develop against your code locally if you put the database in the same location on every machine. The .NET convention is to use the app_data folder in your project's root folder. If you place your database in this folder on each machine you're developing on then you won't need to change the connection string.
For example, copy LoginStuff.mdf into your project's /app_data folder (create the folder if it doesn't exist). Change your connection string to:
<connectionString>
<add name="loginStuff" connectionstring="data source=.\SQLEXPRESS;Integrated Security=SSPI;AttachDBFilename=|DataDirectory|LoginStuff.mdf;User Instance=true"
</connectionString>
Then in code you can reference it by adding a using to System.Configuration and then referencing the connection string:
var conn = new SqlConnection(ConfigurationManager.ConnectionStrings["loginStuff"].ConnectionString);
Between Environments
Between environments (debug, release, etc) you can use configuration transforms to modify the web.config automatically. For each environment you need to create a web.{environment}.config file in your project's root directory. Use XML transformations to specify how you want to modify your configuration file.
These should cover your two most common scenarios. Cheers!