Lyra could be very determined when she set her mind on something. When there was something she wanted, she was bound and determined to get what it was she wanted. And right now, she definitely wanted to find the eggs Fury had hidden. She was curious to see what they would hold, what things she would find inside.
And the only way she could do that was to find said eggs. She was pleased to discover that Fury hadn't hidden the eggs in obvious places. It gave her a chance to think, to figure out where he would have hidden them. She tried to think like him, to figure out where he thought she would or would not look.
Pan seemed intrigued as well. He seemed to forget himself, too, because he called out "Lyra! Over there." He rarely spoke around Fury, but he was getting into the egg hunt as well. And it wasn't that he didn't trust Fury, he was just accustomed to only speaking around Lyra.