If a bounty is offered on a question, the user offering the bounty is asked to select a reason for offering the bounty.
This reason should be displayed or at least available alongside the answer that is ultimately awarded the bounty, if any such answer exists, just like it is displayed alongside the question while the bounty is active. It could be displayed alongside the answer text, or hidden in the tooltip of the bounty rep gain symbol, but there should be some way to get this information.
The reason for this would be that otherwise, unless someone copies the bounty reason into an answer (or in some cases, comment) there is no way to tell what the bounty rep gain was actually for.
Perhaps a good example is my question on SuperUser Why do some RJ45 plugs have the lock release clip under a rubber cover?; the question is currently at a net +77, the accepted answer is a net +109, there's another net +114 answer, but the answer that was awarded a bounty is a net +1. (Personally I don't think that answer should have been awarded the bounty, especially with the bounty reason given, but since it wasn't me offering the bounty I figure that's not for me to decide.) Even when the answer that was awarded the bounty is a good answer, something like that can seem very weird. Making the bounty reason available next to the answer that was awarded the bounty would make it clear what the bounty giver (whether the OP or a third party) was looking for.
