Reinvite

Reinvite module: (mis)using Drupal's core function to generate instant login links for long-lost users.

All the answers we need are in the user password submit function:
http://api.drupal.org/api/function/user_pass_submit/4.7

[Update: Newer versions of the attached below are available. Ask.]

The request came in this form, from my co-Collective member Dan, responding to a call from the executive director of Amazing Things Arts Center:

he said some people forgot what their username was on the site, and now when they're trying to make new accounts it says "email already registered' blah blah blah...