A couple of sarcastic ones: Amblyramphus holosericeus. Amblyramphus means "blunt bill". Balaenoptera musculus (musculus = "little mouse")
A local name for the extinct Norfolk Island subspecies of the Long-tailed Triller was, 'Java Sparrow'. 'Sparrow', is understandable, but, 'Java'? Also, the current Japanese name for the guinea pig (morumotto) is a transliteration of, 'marmot'. I'm not sure whether that's better or worse than its former Japanese name; 'tenjiku nezumi' (Indian rat).
Goldcrests were historically called Woodcock Pilots because they supposedly rode on the backs of woodcocks during migration.
Why not just use the scientific names? For a time that is exactly what I did, however, before and after leaving the scientific community I learned that very few people knew what I was talking about. So I use common names, even though these can, and do, cause confusion at times. As we all know just about every species has several "common names", often due to geographical distribution, even over small distances. My pet hate is the almost constant changing of names, most of which are meaningless to me! Not that I do not understand that changes must be made as we gain more insight and knowledge - but I still find that having to re-learn the names of animals that I have known for most of my life is more than annoying!
Scientific names change frequently too, though. Especially right now, with all of the new genetic tests helping determine the actual relationships between species. Even some mammal groups are a taxonomic mess.
There is some confusion about the generic name of the snow leopard - is it Panthera or Uncia? It has also been classified in Leo and Felis.
For a long time now, there has been confusion regarding whether the European Greenfinch belongs within the genus, Carduelis, or, Chloris.
It's firmly in Panthera. Much of the historical confusion came from them being a a difficult species to study, and people assuming it much be related to Lynx because of color, etc. Pretty much all cats have been in Felis at some point. This does a good job explaining some of it: Snow Leopards On the species level, Felidae is rather solid; the only main room for change would be the most hard to study small species, like the little ones in South America. Mongooses, though, are a big mess. Another that comes to mind is the brockets.
Elephant shrews, tree shrews, golden moles, and flying lemurs are all rather bad offenders imo. At least flying lemurs are adjacent taxonomically to real lemurs and other primates?
Yeah, binturongs are probably the viverrid best known to the general public (which is saying very little of course), but most “commoners” probably know them better as “bearcats” thanks to bearcats (which is admittedly a catchier name) being the mascot for several universities. Obviously, they aren’t actually bears or cats, but just for the record, Viverridae is closer on the tree of life to Felidae than to Ursidae.
Personally, I’ve always preferred to call Bassariscus astutus “ringtail cats” as opposed to simply “ringtails” even though they certainly aren’t cats because that reduces confusion with the better known ring-tailed lemurs that they (vaguely) resemble.