Some answer here concentrate on Isolated Storage. If no other application need access this data then perhaps Isolated Storage is the answer for you. This link is a blog describing how you Isolated Storage is used in a nutshell.
I suspect this possibly isn't what you want since you have the user select the file.
Certainly in SL3 you can have the user select the file and then you can read it.
However in order to save it again you would have to have to show a "Save File" dialog to the user, to aquire a File Stream you can write to.
In SL4 running OOB with elevated privs you can save files like this directly in the users folder such as My Documents.
Edit
Just to be absolutely clear, you cannot write to a file that was acquired using the OpenFileDialog
only read, you cannot read to a file that as acquired using SaveFileDialog
only Write.