I found an easier way to change the sidebar widget's "Welcome" to whatever you want.
1.) In your WordPress install go to the folder: wordpress/wp-content/plugins/theme-my-login/includes/
2.) Open the file: class-theme-my-login-template.php
3.) Locate the word "Welcome" and replace it with whatever greeting you'd like. For example, replace "Welcome" with "Hello".
4.) Save the edited class-theme-my-login-template.php document to wordpress/wp-content/plugins/theme-my-login/includes/
6.) Refresh the page. Now, instead of "Welcome" it should have "Hello" or whatever you put as your personal greeting.