Requires iOS 5 or later, yes you can now trim or cut a part of the voice memos you recorded on your iphone. In order to edit voice memos on your iphone, follow the procedure below.
First open voice memos, the app icon will look like as shown in the image below.
Under voice memos you will see the list icon on the bottom right, tapping on which you will see list of voice memos
list of voice memos
select a voice memo and open its details by tapping the blue arrow icon on the right and tap trim memo
Now trim the voice memo and listen it as well after trim.
One you are done, tap trim voice memo button
Related: Copy, Backup Voice Memos, Recorded Audio From iPhone To PC or Mac, read here