According to the Japanese Wikipedia article for the series:
Because she has almost completely forgotten how to speak English, she has a complex about her last name "Coppola". When she changed schools, she hid her last name "Coppola" along with the fact that she
couldn't speak English and could speak Japanese. However, she was
quickly exposed by Itou and her friends.
So it seems like she's embarrassed because she has a foreign last name but can't speak English.
Also, on an English blog, someone asked the same question, and a Japanese viewer provided their insight via comments (comment #15, don't think there's a way to permalink it):
Kappore was a joke show in the Japanese feudal age. Coppola has a
similar pronunciation to Kappore (especially in japanese’sound). now
almost Japanese don’t know the meaning of Kappore, but we feel funny
when we listen the word like kappore.
There don't seem to be many English resources on kappore, but you can also search on Youtube for some modern examples.
On the same blog's comments, another Japanese viewer comments (comment #17):
The reason why koppora sounds weird to the Japanese has something to
do with the accents of the 19th century peasants in northeastern
Japan. They used to pronounce “kon-na koto” as “koddara godo”,
“chotto” as “chokkura”, “bikkuri” as “beggura” and so on, which is not
considered very elegant among the white-collar class. And, the family
name Coppola doesn’t sound like British but reminds us of Italian
immigrants, which further highlights the awkwardness of Ana’s phony
“British Style.” That’s probably why Miu always mentions “koppora”
whenever Ana gets a bout of British chauvinism (as in drama CD’s) or
her facade is wearing off.
So there are three different theories for you. Seems like even the Japanese aren't 100% why it sounds so funny.