Around here, the main invasives are garllic mustard, autumn-olive, and japanese knotweed, all of which I use in some way.
I eat a lot of garlic mustard. I eat the leaves when theyre the only green available (but IMO they're pretty pungent and so I either use them as a seasoning (primitive pesto with garlic mustard & ramp leaves!!) or I only eat a few). My favorite part is the younger juicy stalks in mid-spring, they're better tasting than the leaves! I gather as many of them as I can, making sure to gather them before they flower because 1) they get very tough when they flower and 2) the flowers turn into the seeds and, as much as I like garlic mustard, I like the diversity of the forests better. I tried drying out the seeds this summer but they were so small and they got stuck in the basket I gathered them in! The dead stalks are very good for starting a fire, as they make less smoke than grass and they catch better, but they're not that great for after it starts going strong.
I just stuff myself on as many ripe autumn-olive berries as I find, hoping that I don't get sick afterwards I've heard that they're really good for fruit leather, but my main collecting device for them is my mouth and stomach (I really like them!!) I haven't experimented with the wood yet, but I bet it would be good for making a bow for a bow-drill kit.. then again, those aren't so hard to make.
I haven't eaten the japanese knotweed shoots yet (I'm planning on it next spring), but, in the fall, when the plants die back, they store drinkable water in their cells! It's not that much, but you could probably get a good drink out of a whole plant. I also use the cells as containers, but they don't last that long. I haven't tried this, but alot of people I know use a tincture from the roots to prevent Lyme disease.
I just started making beaver-tooth carving tools, but since nutria teeth are similar, it might be better to do more with them instead of beavers, as beaver populations are still bouncing back from extinction. Only problem is, there aren't a lot of nutria around here. I'm currently looking to buy them online.