Recreating a bookmarked conversation is a real hassle (there's no "restore" option, even for moderators), so we should be careful with deletions. They're hard to undo.
Further, a bookmark exists in two places: in the room and in the bookmarker's list of conversations. A room owner should certainly not be able to delete the latter, so these would need to be decoupled.
This answer suggests letting room owners mark a bookmark as private, so only the person who bookmarked it can see it. But that can be too restrictive; the conversation is probably also of interest to the other participants in that conversation, and might be useful to others later. The vast majority of the time, the conversation should remain visible to anybody who navigates there from the bookmarker's list. (I'm not speaking hypothetically; I have done this and I know others have too.)
Further, deletion is too much power to give to room owners, who can sometimes be chosen automatically by the system based on who's most talkative. Talkative != trustworthy. While you could argue that nothing is being deleted, only made harder to find, anybody who's tried digging through the transcript of a busy chat room to find that conversation from last week will tell you that this is not easy.
I have a counter-proposal. Moderators can currently delete a bookmarked conversation but cannot do anything else with it. Let's replace that delete button with a mod menu, like moderators see elsewhere on chat, and provide three options:
The first option, "remove from room list", removes the conversation from view in the room list. It would still be visible to moderators, like deleted chat rooms are. This allows moderators to investigate more easily if there are allegations of abuse involving bookmarks. (If you don't think there can be abuse with bookmarks, consider that the bookmarking user gets to name the conversation. 'Nuff said.)
The second option, "private", marks the conversation private to everybody except the bookmarking user (and moderators). This is a rare case, but there are times when we should let the user retain the bookmark but not let him broadcast it. A user with a currently-active dispute with another user, for example, might need to collect evidence in a form that's easy for moderators or CMs to review later.
The third option, "delete", deletes the conversation entirely. This is what the current button does.
It is possible that we would want to extend these options to the owner of the bookmark -- maybe I should be able to mark my own bookmark private without asking a moderator for help, for instance. We should consider making these options available at creation time, but allowing edits later could cause havoc if a user who curated useful room bookmarks later decides to start playing games with access. Bookmarks can't currently be edited after creation, not even to change their names, so keeping that rule (except for moderators as described here) would be consistent.