Some fun ideas:
1)Eugene Schieffelin never gets the insane idea to release every bird in Shakespeare's plays in Central Park. So, no European Starlings in North America, but, he's already released house finches, and a number of other birds that just ended up dead on the sidewalk come October. Instead, Monk Parrots, Finches, and Budgerigars (Some other Romantic idiot decides to release a few hundred in NYC as others escape and develop breeding populations in the US starting in the 1920's) become some of the most problematic, yet useful Invasive birds, kinda like OTL, although Monk parrots are more encouraged. Avian Biodiversity is up massively over OTL.
2)Arachis glabrata (Aka Peanut Grass, or Perenial Peanut) becomes a common type of ground cover. Of course, this has it's drawbacks, as it's fairly hard to kill without modern pesticides, and within the right climactic conditions, quite a potent weed, like Bamboo, or Dandelions.
3) Someone decides to set up a Feather industry on the Iberian Peninsula, and starts raising Carolina Parakeets, Peafowl, and other valuable sorts of Birds, before going bankrupt, and letting the birds escape, just as Cockleburs start popping up as weeds in Europe (Carolina Parakeets were a useful biological control for those, and weren't all that destructive otherwise.) It turns out that intensive farming did a couple of things to the Parakeets, and rendered them fairly resistant to Poultry Diseases (That which most likely finally wiped them out IOTL.)
4) Some idiot (probably an Ex-Confederate) releases a couple dozen breeding pairs of Ivory Billed Woodpeckers in California, Tasmania and New Zealand in the 1870's. They get themselves established. Hilarious Irony Ensues.
Does anyone else have Kim Todd's Tinkering With Eden? It's a good overview of some of the odder stories behind Some of the Invasive species in the US and the Characters who introduced them (Like Eugene Schieffelin, and Captain Tooey,) also her What if on Kangaroos is pretty good.