For example in the sentence:
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The meanings are quite similar, but there's a subtle difference.
Both phrases involve settling for something, or one person making a choice on behalf of another:
お昼ごはんはハンバーガーでいい？ "Would a hamburger do for lunch?" Or something is sufficient:
そこにおくだけでいいです "Please just put it there [and I'll do the rest.]"
でもいい has an added sense that there is something better or preferable out there, but you are settling, or asking, for something less. This comes from the use of
でも, which has a negative conjunction.
お昼ごはんはハンバーガーでもいい? has a nuance of, "I know you like something else," or perhaps, "I don't know if you like a hamburger or not." Another example would be:
名前を書くだけでもいいですか？ and this would mean something like, "[You want to me to put down my address for this petition, but] would it be OK if I just write my name?"
I cannot comment yet, but I would just like to add a sidenote to the existing answer by Kawaguchi-san
There is a site that I use for my studies now, and recently I found a very helpful article there on the difference between でいい and がいい
I just thought that this might be a good addition/expansion on the topic
P.S. Also, to understand the nuance, I personally find it helpful to translate も in でもいい as ”even", e.g. これだけでもいいですか？ ＞ Is it OK even if I do just this?
Though, if you draw a parallel with the 'usual' function of も in general and, especially, も/は for amounts (e.g. 百人もいました > There were as many as a hundred people), then a 'more universal' translation would be "as much/as many as", so the example about would translate as "Is it OK if I do just as much as this?"
Interesting, I have not thought about it before I started writing this