Call ahead. Let your relatives know that nothing personal is intended by her rejection. Encourage them to give her some space when she enters the room and not immediately make a mad dash to grab her. She'll need time to warm up to them.
Give her home-court advantage. Your shrinking violet will be most comfortable at home, but if you're going to Grandma's house, try to make the environment as familiar as possible. Bring her favorite toy from home and play together. Give Nana the toy and see if your shy one will also play with her.
Keep in touch. Don't hand her over and disappear. Stay close by and within her line of vision. She'll feel safer if she's able to see or feel you.
Adjust your attitude. If you're tense, she will pick up on it and feel anxious, too. Of course you feel terrible that she is so frightened, and sad for your loved ones who have been eagerly anticipating her arrival, but she'll do much better if you wipe that worried look off your face and smile.