How can I help my niece pronounce words better, without igniting in her boredom or resentment

Question

for example, she'll say things like "brudder", instead of "brother". She's six.

Answer 1

My son is almost five and has been in speech for two years, I was taught to model the word back to him correctly. I do this constantly with him as he mispronounces many words, your like a tape recorder only your saying the word properly.

Answer 2

You don't want to make it too obvious you're correcting her because you may make her self conscious about it.  At her age, mispronouncing some words is normal.  

Simply repeat what she says and say it correctly.  Like if she said, "My brudder and I played outside."  You say, "Oh, you and your brother played?"  

Chances are, she will self correct.

Answer 3

when she says things like this correct her with the correct way to say it she may catch on