Interesting.. That should allow you to edit.
we'll just do a revision of steps again just to make sure:
1. Go to Security Role Manager of the Role.
2. Verify if the user has 'read' on resource Type - select Settings from View dropdown label, Select settings -> Notification Server -> Resource and DataClass Settings -> Resource Types -> Asset Types -> IT -> Computer. Check that user has 'Read' and 'Create Resource Instance' permissions on this resource type.
3. Select if the user 'Edit' permission in the right click menu - Settings -> Notification Server -> Right Click Menu -> Edit (there are two edit's here). Ensure user has 'read' permission on both of them.
4. As you can see computer and also edit item action, first two steps should be ok with you as well. May be you're something in here ---- Select Resources from the View dropdown Label. Select Organizational Views ->Default -> Ensure user has read, write, create children permission on this group.
4.1 -- Once this is done, you can also click on Asset -> Network resource -> Computer and check if the permissions given above are inherited from the parent ( Computer should have Read, write, Create Children permissions). if not, you've to inherit these permissions from parent by clicking on 'advanced' button at the bottom, untick 'Inherit permisisons entries from the parent object that apply to child objects) and save changes. choose copy from prompt that shows up. Now you can add permissions manually (read, write, create children).
And this should defintely allow you to get edit control on the resource instance.
In case, you don't want to choose default org group and give only control to the instances they've created- make sure the computer instances are in the Org group they have access to.
Hope this helps.
-Prasanna.