Since Drupal 6 has reached end-of-life, I decided to dive in and upgrade the tech blog to 7.5. It wasn't nearly as smooth as all the minor updates, but I think I've managed to work out most of the kinks.
First off, the update.php script was throwing loads of "module not found" errors. Not sure why, but it doesn't seem to affect the actual performance of the site.
The next problem I had was due to my kludgy hack of the default "Garland" theme, which I fixed by resetting my custom logo selection under Appearance, then reloading the logo.
Third problem I encountered was a "Recoverable Error" on line 2004 of theme.inc where a call to drupal_attributes() was being passed a null instead of an array.
Line 2004 appears below:
$output = '<table' . drupal_attributes($attributes) . ">\n";
After looking around the Drupal help pages, I found a reference to a bug that was seemed similar, so I tried inserting the following line just before line 2004, and it seems to have fixed the problem.
if (!is_set($attributes)) $attributes = array();
After updating to 7.52, this same problem recurred. This time I tracked the problem down the same way, and made a similar modification the the theme.inc file. I still have no idea why this is ocurring. The problem appeared to occur because of a missing custom favicon, but I fixed that and it's still giving me the same error. Definitely not the most elegant fix. Oh well. Perhaps the ultimate solution is to ditch Drupal altogether and migrate the tech blog over to Laravel.