Well, you can always hit the selection CheckBox, one at a time, and execute the download option for only that message, modifying the file name to make each message on the same day unique. (For those who have not noticed, there are two download options. The first is below and separated from the list of messages in the Inbox (or Outbox or other folder) and says Download all Private Messages as :, below which is a hyperlink to select XML, CSV, or Text.
The second is to click on the box at the bottom of the list of messages, choosing “Download as XML/CSV/TXT” and clicking “Go”. This option downloads (or otherwise affects) only those messages that have the CheckBox filled.)
If you want to move 60 messages, of course, that might take a while.
You could use the mass download, specifying the .csv option. Each message is given its own row in the table with the columns Date/Time, (originating) Folder, Title, From, To, and the Message. If you need a separate file for each message, you could then open the .csv file, highlight and copy each row, then paste it into a new file that you create. (You can do the same thing using a .txt file, but the messages are separated by a string of equal signs.) This would actually take longer than the first suggestion, but would allow the download to be out of the way so you could play with the messages on your computer.
If you are handy with some editing software, you could download all the messages at one time, then run a program to read the file, breaking it out into separate files, one for each row (.csv) or one every time you encounter a line containing all equal signs (.txt).
I don’t have experience messing with XML, so I don’t know whether it has an option to create separate files by name during the download.