Some time back, there was a discussion on the list about the labels of collapsed branches. As it is now, we display a bit of text from the branch. The general view was that this is not very useful, and it makes documents with lots of branches look terrible. The proposal was to label collapsed branches the same way we label uncollapsed ones: with the branch name, and a marker showing whether it is "on". Info on what's in the inset will still be available from the tooltip.
I've attached a patch to the bug report http://www.lyx.org/trac/ticket/10213 and intend to commit unless there are objections. Richard