Trying to eliminate easy-fix possibilities, have you tried swapping the AV-out cable with your AV Famicom?
The adapter I'm running on my SNES-Mini is model SNS-002 - 10V 850mA
For a boost in quality, you could get the SNES mini modded for S-Video & RGB, tho that may be a can of worms you don't want to open on a potentially malfunctioning console. On the other hand, having an experienced modder look at it, it might be an easy fix and an awesome upgrade.
There's a definite increase in the picture clarity between my modded mini & the original model I have from all the output signals.
Just a stab in the dark, but it may be a capacitor problem. There's a row of electrolytic capacitors on the mobo.
If you're familiar with a multimeter, and
purchase or
make a Security (game) bit, you could test those yourself, and find out:
http://console5.com/wiki/SNEShttp://www.learningaboutelectronics.com ... -capacitor