I think Lexie, Linny and Lisa have good points. Different children are going to react to that circumstance differently and the way that you parent affects how your children cope.
I look at it this way...
If there's only one of you and few people have any experiences with people like you, then theoretically there are fewer expectations of how you're supposed to behave and you are free to be who you really want to be. In practice, tho, it's really not so freeing to carry the mantle for an entire race/ethnicity and set the standard by which others will be judged.
What's more common is that there are very few of you close-by but there are many others just out of town (of whom people may or may not have favorable impressions). In this case, you may be unneccessarily constrained (expected to do/be as they are) since that's all that people know. To be an individual means you may become estranged from the only people who look like you. To be as they are may mean not being true to yourself. Not a fun choice to have to make.
The latter was the situation I was faced with and it was a very painful time in my life (there were fewer than 10 blacks in a graduating class of 300+). Dating was a challenge. Being smart was a challenge. Not being into cheerleading was a challenge (jk).
Yep...plain ol' challenging.