What should a parent do to intervene in a relationship with an important relative and a child? For example, if it is readily apparent that your child doesn't like her grandmother, even if the child doesn't verbalize it what should the parent do?
What if a relative (the same or another) doesn't like your child?
How do you foster the idea that a child must not allow her feelings to create a lack of respect for the relative, for example when the relative is leaving the house the child must say goodbye in an appropriate way.
IMHO Peter is overthinking this.
Don't worry about feelings, worry about behavior. Insist that your child behave properly, and if someone is behaving poorly to your child, let your child know that it isn't their fault and that they still need to behave properly.
1 - Child doesn't like someone else
Do not try to manage your child's feelings. All you should expect from child is to treat others with courtesy and respect. They don't have to like crazy Aunt June, but the dislike should not be obvious.
However, talk to your child and probe into what is going on. It may be silly ("She smells funny"), or it may be chilling ("She yells at me"). Do not take your child's concerns lightly, and do not force your child into uncomfortable situations beyond common courtesy.
2 - Someone else doesn't like your child.
Obviously, you can't do anything about what crazy Aunt June thinks of your child, unless your kid is obnoxious and Aunt June is right. What you need to do however, is work with your child on the idea that not everyone is going to like them, and that is perfectly OK.
3 - Common Courtesy
This is actually the tough part ... you must continually impress upon your child the concept that everyone is to be treated with courtesy and respect, and you must teach what that means in terms of behavior. Whether they like the person, or the person likes them, or the person treats them poorly, the person is to be treated with courtesy and respect.
I am afraid courtesy is a dying concept, and that is a shame.
A lot depends on the details. Who is the relative, what is his/her relationship with the parents and/or other family members? Some families may have hidden, suppressed - or even open - tensions or conflicts between members. E.g. a mother-in-law may - secretly or openly - feel that she has been robbed of her son by her daughter-in-law. Etc, etc. Children can sense such tensions even if adults are hush-hush about these, and react upon them, even unconsciously. If there are such issues in the family, raising them to the light and resolving them properly may eventually solve the child's symptoms too.
In general, I believe the best approach is to accept it. We may try to talk with the child about why (s)he dislikes the adult in question, but - depending on her age - we may not get much useful information out of her. The person in question is also better accept it and keep the (physical / emotional / social) distance set up by the child. Follow her, be relaxed and be ready to accept any future communication attempt, but not force contact.
The parents may want to ensure some minimum level of politeness and social contact, such as that the child must greet everyone properly, regardless of likes and dislikes. But that, again, is case specific. When I was a kid, there were adults in our family and among my parents' friends whom I plainly refused to greet, or get in contact with. My parents accepted the situation then. I still managed to learn the value of greeting people since then, and nowadays I greet everyone sincerely, out of my own will, not just because it is "polite".
I think we should try to get to a healthy balance here. Obviously, raising a child to be totally unaware of social conventions is not good. At the opposite extreme, forcing him/her to be always polite and nice with everyone, effectively making her totally repress her internal feelings and thoughts, is not good either. Fair enough, every society is doing this to its members to a lesser or greater extent, so we can't fully avoid it, just strive to find the right balance between national / family culture, ourselves and the child in question.
Well that's a sensitive one. My feeling is, if that relative is important to you, and you have a good relationship with him/her, (s)he will probably more understanding. If she is not understanding, and reacts in an irresponsible way to your child, you may prefer keeping a distance to her anyway. I think in some cases the least worst option is just to try to minimize the occasions when you (and/or your child) need to be with such a relative.
This depends on the child's age - a younger child acts purely out of feeling, and has no concept of "respect", much less "social conventions" or "dignity". To such a child, I would just say that we greet people when we meet or depart, period. Even those adults whom we may not particularly like. Now this is a sticky issue, because children usually are very sensitive to any discrepancy between our words and our own inner feelings. So I would definitely not pretend liking that relative if that is not the case. However, explicitly mentioning in the company of a smaller child that I don't like aunt Maggie is risky too, because I may end up hearing my exact words shouted out in the middle of the next great family reunion :-(
So the best approach may be to consistently stick to a rule, and explain only as much of the background as necessary for the kid (if (s)he is specifically asking about it), on the appropriate level. Actually, in my experience, they often are very understanding on a subconscious level and don't ask those embarrassing questions openly.