What do you mean by "under construction mode"? Is the site live, or is there only a placeholder there for it?
Remmber that just becuase something isn't reciving much traffic, that doesn't mean that the server will handle it. Some servers are terribly under-powered. Shared hosting platforms can have issues with other sites using too many resources.
Have you looked at the error log for your site? As well as that there's also the MySQL slow query log. There's a few places like that that you realyl should look at to find out what the actual error messages are. The log files will point you to the right direction to find out what's happening and see if it can be fixed.