I'm currently building a WordPress theme with a Theme Options page. I used the original WordPress Theme (Twenty Twelve) as a basis and just adjusted and added as I went along. The problem I ran into is that the page does seem to save my default settings, but when I change them in the form and hit the save button, it still only pulls up the defaults.
This is my theme-options.php, originally in Pastebin but the link has expired, that is included in my functions.php.
You're not consistent with your naming, change fatum_box_one in your options array to eerste_box, etc., and change the names of your textareas from fatum_box_one[eerste_box] to fatum_theme_options[eerste_box], etc..