I'm missing my core search module
I'd guess that your custom module is called search, which would conflict with the core Search module, also named search.

There's no such thing as namespaces with regards to Drupal modules (at least out of the box, I'm not sure if there are any contrib modules that deal with this. My gut instinct is that there wouldn't be). Each module needs to be individually named, or the last module of the same name that Drupal comes across will be the one it uses.

Make sure your module is not called search; that includes making sure the folder name for the module, the name of the .info file, name of .module file, etc. are not called search.info etc. That should make sure that both modules are available in your module list.