Well, Stream works by connecting to your WordPress.com account. If you connected it using your own account, you can reset the data by disconnecting, then reconnecting to your clients WordPress.com account. That would keep Stream logging (which is very handy if the client breaks something), but not show any of your previous history.
If you were to reconnect with your own account later on, your previous history would be shown, but not the clients. Keep in mind, though, that Stream only logs 3 months worth of data.
All this advice will be changing shortly, though. So stay tuned.