I have a 2010 Standard Farm. Over the last few months of setting up our new intranet, lots of accounts were added to the root site permissions. Most of our site collections inherited those accounts, including "SHAREPOINT\System". It was deleted without fully understanding why it was there. Since then, I reset the search index. Now I can't get it to crawl any of our sites. Was this account added to the root site collection so it can read in the sites? Is it as simple as adding it back in? Or is this issue completely unrelated?
The search crawl account should be given special read permissions via web application policy in Central Administration, not by having an entry in the site. It should also not be the System account as using that account to index your site will include a lot of noise in the results. Adding the Web Application Policy is normally handled automatically when the crawl account is defined but it is possible that it was deleted so check there first to make sure it still has read permission to that web application.
The next step is to make sure that the content source in the Search Service application is configured properly to either include all local sites or include your site by URL. It is also a good idea to make sure that your Alternate Access Mappings are configured properly.
If any of the above was out of order, correct that, reset your search index and kick off a full crawl.
To answer your original question though, in a properly configured farm there is no reason why the SharePoint\System account should ever appear in a site collection's permissions.