On the menu-customize page, after
1) dragging a menu item, or
2) checking/unchecking an enable checkbox,
and clicking “Save configuration”, changes to menu were not saved in Drupal 6. Hence, I started to debug the issue at the Application level. After long head bang, I raised my hand and took some rest. Finally, I googled and found that the problem was not at the Application Level but at System Level. I came to know that while posting the form (i.e. when Save Configuration is clicked) the data sent was of large amount and the system was not able to accept since maximum input size was not defined. Following were the changes made in php.ini file.
; How many GET/POST/COOKIE input variables may be accepted max_input_vars = 2048
After making above changes I restarted the Apache server. Thus, the problem was resolved and Save Configuration in Menu started working.