Why is the content of existing blocks not editable?
I suspect that the blocks that are doing this are made via hook_block and not with the block UI (ie, admin/build/block/add).  I just checked a Drupal 6 site that I did a while ago, and all of my custom blocks have this behavior.

The Configure link for each custom block should contain the module name that defined it, along with the delta.  For example the "Who's online" block has this for the configure link:


  admin/build/block/configure/user/3


It is defined in user.module and if you look at user_block you will see that the $delta for this is 3.