She liked this man. She heard Jacob in her head, warning. Emmett, father, cautioning. But Renesmee also felt she could detect certain things, especially danger. And while this man was dangerous, he wasn't to her at the moment.
Renesmee took the offered hand and shared more. She showed him a glimpse of the confrontation with the Elders of her kind, sent to kill her for what she was. She showed all the vampires she had had to convince that she was peaceful, one by one. Renesmee knew she unique and different. She showed him her family, all their different gifts. And the boy from the Amazon that was just like her.
Lastly she showed him an image of Jacob, her werewolf friend and soul mate. That was a bit of a long story, how that came to be but that wasn't the point. She showed them playing together, of Rosalie sniffing and making a comment about the 'dog'. She flooded him with the full happiness of the situation. How happy she had been to be there.
The images stopped again. Renesmee was always honest and her images made her feel freer and she shared them openly with the stranger. Later, she sent a short image of him, the way he looked when she saw him across from her. It had an air of wonder. Did he have a name?