I have a Drupal Menu which I created through the interface. I wish to add an entry which says
Hi, John Doe
Where "John Doe" is a link to the User's profile page. I would like to do this programmatically or if it can be done through the interface then that would be great.
You cannot add HTML-ified items to the menu. Menu-items are always wrapped in an anchor tag and will filter out HTML. Changing this behaviour is technically possible, but will cause a lot of unknown side-effects.
What you want, is either a simple theme override, or a custom block (potentially without a title and with some other bullets).
You can create a block with hook_block or by simply typing the HTML, with some PHP into a new block, using the PHP-input filter. This last option is quick, but is discouraged by many people for performance and "good practices" reasons: you should not store php in your database.
EDIT: After comment about "just the name":
To insert "just the name" you simply need to create module with a hook_menu.
Is about what you want.