If your main limitation is budget, then I'd say eggs and beans/lentils. While beans aren't particularly "quick", you can cook a big batch of them and freeze them in smaller portions. Lentils can be made fairly quickly though.
Nuts are great but may be a bit expensive, and I don't agree with soy at all.
Check out your whole grains. Protein varies from almost none to lots.
For breakfasts, there are lots of different ways to prepare eggs to keep things interesting. Eggs can also serve for lunch as egg salad, deviled eggs or even just hardcooked. Or you can make bean dips like hummus for veggies, or nut butter dips for hard fruits. Don't forget mung bean sprouts, which have a fair amount of protein in them and are great for eating straight or tossed with a light dressing or topping a sandwich/salad.
And it looks like your dark green veggies have a bit too - broccoli, brussel sprouts, kale, collards, peas, dandelion. They're in the neighborhood of 5g per 1 c cooked, but it's still something.
You say they don't tolerate dairy very well... can they handle yogurt (or kefir)? If they can, incorporating yogurt into a breakfast smoothie or making a dip like tzatziki for veggies can give a boost to their protein intake, too. I always drain my yogurt really well, and that gives it more protein per fluid ounce since you're removing the water.