Thanks for the time you've taken on this - much appreciated.

The best you can do is what you have already done, which is to include the Page in the Navbar and then require a password to enter it.

Alternately, don't put it in the NavBar/Menu at all and only give the address out to the project workers to bookmark. I would still password protect the Page.

For the sake of this experiment, would you please remove any Custom Menu you may have set up. You don't have to delete the menu, just remove it from being the Primary Menu in the "Theme Locations" Module of the Custom Menu Dashboard.

I want to hide that menu item from people who are not logged into the site.

Please check the Publish module for that Page to make sure that the visibility is set to "Private". If not, make the change and update the Page.

What I do get when I click on "Project Area" is a place to enter a password, which to me means it hasn't been published as "Private".

@raincoaster.
Makes sense - but it's not what i'm looking for.

@justjennifer
There's only one private page at the moment: >> Project Area >> "Stuff for your eyes only" :)

If I log in and go there, there's a page visible. If *you* go to that menu item you get a "404 not found." I want to hide that menu item from people, like you, who are not logged in.

Having the private pages is good - but I don't want users to have to remember the URL - I'd like it to appear on a menu somewhere.

http://en.support.wordpress.com/pages/page-visibility/
http://en.support.wordpress.com/posts/post-visibility/

Which of your Private pages are still visible when logged out?

If I add a private page to a menu it works for logged in users and gives a "404 not found" for the public. This is correct but ugly.

Is it possible in wordpress.com to have the private pages appear on a menu to logged in users, and have the menu item hidden from the general public?

My first problem is not to be able to delete the "Blog" -looks-like-a-button but i guess is a page- on the left lower corner of the K2 Lite theme. It overlaps my customized header.

A custom menu should help with that. You can create a custom menu using the Appearance → Menus page in your blog dashboard. Make sure you set the "Primary Navigation" option on that page. I checked your Appearance → Menus page, and it looks like you created four separate custom menus but you didn't set any of them as primary.

Here is a help page with details and visual examples showing how custom menus work:
http://en.support.wordpress.com/menus/

If I add a new page, a button-menu will appear next to it which i can delete but i cannot delete this one blog which cam with the theme. It was containing the default hello world! post. So i start to believe that in this theme maybe i cannot modify that. I searched in the forums but without luck.

I am ready to migrate to another theme or create my own theme if necessary to solve this. I do not find the option to start a theme from scracht and then shift my blog to it.

Thanks for the help