Although you don't specify this in your question, I assume that you're going to develop the new version of your application/database, and then at some switchover point you need to migrate all of the live data from your old database into your new database.
If this is the case, then you're really asking about two distinct processes: the migration (with some modifications) of the database structure, followed later by the migration of the data itself.
For the first process, the best tool is you, the developer (I don't mean you're a "tool" - you know what I mean). You could bring over the structure of the old database and then change it as necessary for the new version; however, this approach in general tends to leave too much of the old structure behind. I think it's better to take advantage of the situation and rebuild the database from the ground up, using the original database just as a general reference.
For the second process, I would treat the data migration as a separate task requiring a separately-written and -tested application. This application could be a set of scripts or a compiled application or whatever is most convenient for you. Because your old and new databases will not have the same structure (and may in fact be very different), there are no commercial tools out there that will handle this task for you automagically. By treating this as a distinct application that you write yourself, you can test the data conversion process many times before your "go live" date.