First thing to try with the mini is a new HDMI cable. That might fix it.
The Genie Mini has ALL channels protected by HDCP, not just the premium channels. If you want to try using Component Video instead, you need a special "dongle" cable. Call DirecTV and they might send you one for free. Or you can order one from dealers like Solid Signal ....
REPLACE THE HDMI FOR 10pin Mini-DIN CABLE
High-bandwidth Digital Content Protection (HDCP) is a technology that protects programs from being copied from an HDMI or DVI output to a computer or other digital recording device – like a DVD ******, etc...
You will need to replace the HDMI cable for a Component one, but since you're using a GENIE mini you won't be able to conect a regular componet cable, but if get a 10pin Mini-DIN cable (http://www.google.com/imgres?imgurl=http%3A%2F%2Fpimages.solidsignal.com%2FH25-10PIN_2_zoom.jpg&imgrefurl=http%3A%2F%2Fw…) and I asure you this will get your issue resolved and you will keep same picture quality.
Remember: REPLACE THE HDMI FOR 10pin Mini-DIN CABLE
You can get it from an locaI store, or you can call Directv in order to get it!!