Hi! Come and check out this thread for lots of ideas on self-feeding:
http://babywhispererforums.com/index.php?topic=224254.0Bananas and avocado are both great - what about other fruits like steamed or roasted apple (cut into wedges or crescent or circle shapes), pear (raw if very ripe, or steamed), watermelon, mandarin segments. For vegetables, how about broccoli florets (steamed just soft enough to mush in your mouth, but so they still keep their shape)? At 9 months, my LO loved pasta with sauce (a shape like spirals or maccaroni is easy to pick up), or rice with sauce. Meatballs, fishcakes, fritters, etc are all good things to offer that LOs can pick up easily. You want it to be big enough so they can hold it in their fist and still get at one bit sticking out, if that makes sense!
It's not a problem for some of the food to come through undigested - that's still the case for my LO at 12 months. Maybe try giving BIGGER pieces, rather than smaller, so he really has to chew with his gums and mush it to be able to swallow it!