Then they could reply "I like that idea!", done. Adding the ability to like an item in a conversation is just wasted database space, in my opinion. I could see if like 100-1,000 people wanted it - but judging by the posts here, that's not the case.
Conversations in general do not need to be liked. To 'like' something is to express an interest and for entire community to recognize. If it is private, there is no point, as they will not be able to view it?
Imagine your recent activity: Audentio recently liked a private conversation. (No link to view it as it is private) No point.
Well... like applies to community and since it is counted in a displayed total for each person on the board...and is viewable publicly, so should be the posts which have gained those likes.
liking msgs in your inbox would kind of open up circumventing the transparency of the like system the way it is now and more than likely would be abused.
On the other hand I have often gotten help in an inbox msg and really wanted to like the post for the response before I even followed the advice in the post just as a way of saying thanks quickly as I continue to read their post.
The only way to meet halfway is to be able to like a inbox msg and have the like be null as far as adding to the user's stats that can be seen publicly.