Could someone create a plugin which would allow people to add links to the 'user name' drop down menu?

You mean this menu?

Should you, as the admin be able to add links? Or the users? What links would you put there, and why can't they be added to the normal menu?

I don't know what OP wants to put there but for example I recently wanted to put a link to billing/shipping address settings there. 

I'll share what I've done.

AdminCP > Customise > Skin Name, Under templates > Core > Front > Global > userBar

Or simply search in templates: ipsMenu_item

Find:
 

<li class='ipsMenu_title'>{lang="menu_settings_title"}</li>

Add below:

<li class='ipsMenu_item'><a href='URL TO LINK'>{lang="NAME OF LINK"}</a></li>

I'm not sure if this is 100% correct, and I'm sure someone with coding experience could confirm. Remember to change the URL and the NAME OF THE LINK.

But this will add extra links to the drop down menu.