First things first:

• Zeitgeist is essentially fashion for ideas. See a discussion about it on Wikipedia.
• Why the name ? Because when this idea was first discussed, someone on Drupal (chx, I think) said 'Oooh, a Zeitgeist', and I liked it.

• Log searches with their timestamp
• A customizable drupal block with the 'n' most recent searches. Since v1.4, 'n' is settable
• A customizable drupal block with the 'm' most popular node searches during the current month. Since v1.5, 'm' is settable
• 4.7RC2 compatible.

• A page displaying the most requested searches over a given time span
• A RSS feed tracking all searches
• A periodic (weekly) summary of the top queries, by email to the site admin
• A setting to include queries by UID=1 or not
• make the customizable blocks themeable, not just customizable

Looking further:

• ability to log various site uses, instead of just searches.
• coupling with statistics.module

• Table 'zeitgeist' contains the search log
• Variables:
  • zeitgeist_history: the duration entries are kept in the log table
  • zeitgeist_latest_count: the number of entries to be displayed in the “recent searches” block
  • zeitgeist_latest_info; the customized info of the “recent searches” block (in the blocks list)
  • zeitgeist_latest_title: the customized title of the “recent searches” block (when displayed)
  • In zeitgeist_latest_info and zeitgeist_latest_title, the string %count can be used to insert the value of zeitgeist_latest_count

As of 4.7, Drupal does not implement a standard logging API, be it for search or anything else. The net result is that code like zeitgeist.module, which needs to record data from core, must obtain them from undocumented behaviours.

• The core patch from which this module has been born used to parse the standard “watchdog” table.
• For zeitgeist.module 1.1 to 1.6, this meant parsing both the “search form” form and search URL, which led to various problems, notably regarding potential safety issues, and the algorithm proved unreliable over minor core releases changing search.module.
• Version 1.7 only processes the “search form” form. This simplifies the code, but leave it vulnerable to search.module changes.

At some point, drupal core will probably evolve to include such a logging mechanism, and zeitgeist will then be able to use it. Should the module become popular, it is also conceivable that search.module add a recording hook, independent of other recording needs.