Veggie burgers - I make a batch from Mark Bittman's cookbook, and keep a few in the freezer at all times! My daughter loves the beet-almond-bulgur burgers.

Make sushi rice and form it into little balls. DD loves this... she does tend to smoosh the balls in her hands, but it sticks together enough that she can still get it in her mouth.
Cheese cubes, always a huge hit. You can do softer cheese like mozz if you think the texture might be too much.
Edamame and beans, PP's were spot on. Also soft-steamed cauliflower (smoother texture than broccoli) or bits of potato.
Why can't you send some yogurt or applesauce? He needs to learn to use a spoon at some point...

Just chop up whatever you are eating, nothing to it.

thawed peas (no need to cook them!)
well-cooked edamame
tofu

Spinach souffle
Beans
Quiche
Mostly anything cut up small

-pita bread cut into tiny pieces with hummus on them
-black beans or any other kind of bean, cooked soft
-can he eat egg? If so, either scrambled egg pieces or hard boiled egg cut into small pieces. also, french toast cut into small pieces.

I really need suggestions for finger foods besides pasta that are healthy options to feed my 13 month old. He's very insistent on self feeding, and I normally do chopped up fruits, avocado, pasta, cut up chicken, but I need dinner/lunch ideas. (I usually send DS dinner left overs to daycare for lunch the next day). It needs to be somewhat soft as he's still learning to chew, eat, etc.

Thank you in advance, I really appreciate it!