After reading your post, I've been playing around with my copy of FMP 10adv and here are my findings:
If you ungroup your web viewer--which removes the button setup--your object name re-appears.
If you select another layout object, such as a field, name it and then format it as a button, you see the same behavior--its object name disappears.
If you format an object as a button and then name it, the object name will be visible in the object info palette. If you then ungroup it, the object name is lost.
This may be counter intuitive at first glance but I think this is more a limitation of Filemaker rather than a bug. Current versions of Filemaker are unable to display the object names of any object that is part of a group. It can only display the object name of the entire group. To see the object name of a group member, you have to ungroup it.
Obviously, this limitation can be a significant nuisance since ungrouping a button destroys whatever action was linked to it.
Any one care to join me in using the suggest a feature form to drop a request into FMP inc.'s wishing well? (Go to company | Contacts | Suggest a feature.)