I’d target confusing or misleading names. I’d rename red-bellied woodpecker to “red-capped woodpecker”. And the name for Connecticut warbler would have to go too, not sure what I’d rename it.
Many plants already have a lot of common names and you can just choose to use one over the other. For example, I think “Spreading thistle” is a better name than “Canada thistle” and I think “Japanese creeper” is a better name than “Boston ivy”, because these plants are not native to Canada and Boston, respectively, but the second one is native to Japan. One thing I like about iNaturalist is that it often seems to use the “better” name even in the cases like with these species, where the more misleading name is in widespread use.
If you gave me broad leeway I’d probably ditch “honorary” species names and replace them with more descriptive name, especially in the cases where you have many ones named after the same person, e.g. Swainson’s thrush, Swainson’s warbler, Swainson’s hawk.
These honorary names bother me for several reasons. One, they’re not descriptive and thus they make it harder to learn and remember the species, because, unlike most common names, they don’t contain any useful information about ID, habitat, or range. Two, they’re almost always named for western people, and as such I think they are kind of monuments to western colonialism/imperialism and perhaps also classism. Like, no offense to William John Swainson, he’s an important English ornithologist and naturalist, and he also overcame some interesting obstacles in life, and is kinda a fascinating guy, and I see the desire to recognize him. But at the same time, we western europeans (I am mostly of western European heritage) did so much to erase and destroy Native American culture, some intentionally and some unintentionally, and it just seems wrong on a moral/ethical level for us to be honoring people from our own culture, on a continent where we so thoroughly eradicated the native cultures. If we’re going to honor anyone at all in the naming of species, I’d rather it be in the form of using names for these species from Native American languages, for the languages indigenous to the same ranges as these species, to honor those people. And, even setting aside any of the issues with native American people, I don’t like the idea of honorific names because they always seem tainted by classism: it’s usually people of higher socioeconomic status who get into positions of prominence in science, and Swainson is certainly no exception. Since I believe in the inherent worth of all human beings, I dislike these honorary names on ideological grounds.