I hate her. She's always treated me so badly that......
a. I'm even reluctant to mention her name.
b. I'm reluctant to even mention her name.
They are both correct but the meanings are slightly different.
In (a) "even" modifies the entire phrase, "reluctant to mention her name" it does not modify the adjective "reluctant" alone. Thus the alternatives the speaker has could be (i) meet the person (ii) talk to the person (iii) talk about the person to a stranger (iv) talk about the person to a friend (v) mention the name under any circumstances, etc. From these options, the speaker has chosen the last.
In (b) the word "even" modifies the verb "mention". The implication is that there are several things that the speaker could do with the person's name (i) say it to the person (ii) write it on paper (iii) mention it to a third party (iv) shout it from the rooftops, etc.
The speaker is so nervous about uttering the name under any circumstances that she is reluctant to do so much as even mention it.
So, you can see that "even", in this context, implies that there are several actions possible and that some are worse than others. In (a) the speaker has a choice of ways to interact with her (or people around her). In (b) the speaker has a choice of ways to interact with her name.
Does this help?