Your sentences are all right, but they have slightly different emphasis. Emphasis is always on whatever comes after 'even'. The difference is very subtle.
"I can't remember even her name."
Usually:
- (If emphasis is put on 'her') You can't remember the names of lots of people, but in particular you can't remember HER name.
- (If emphasis is put on 'her name') Of all the things you can't remember about her, you can't remember her name. This is essentially the book's answer.
"I can't even remember her name."
I just spent ten minutes trying to explain this in words, but I'm still struggling to. It can be interpreted so many ways, and I think all of them are pretty intuitive in their context if you're human. XD
I'll just say that it emphasises 'remember' or 'remember her name'.
This is the most common way to say it.
I think the other also pretty intuitive. You just need to look at what comes after 'even'.
For example, "I don't know even the people next door" implies that you don't know a lot of people (including the people next door).
"I don't even know the people next door" is just a statement that you don't know the people next door. It could mean the same as "...even the people next door" in the right context.
Really, a lot of the time you could use either. The answers the book suggested are definitely much more common, but just as natural. If you just want to use 'even' in a sentence to emphasise its meaning, put 'even' before the first non-auxiliary verb. If you want to emphasise a specific part within the sentence, put 'even' before it.
All of the following are possible, but are slightly different.
"Even I can't remember her name."
"I even can't remember her name."
"I can't even remember her name."
"I can't remember even her name".
I don't know if this was helpful at all. I realise that saying "use your intuition" isn't very useful.