We moved just over six months ago. We moved across the country (US), leaving our friends and everything familiar to Ethan. Playing games and Skyping with friends was a good transition for him, but I can tell he's ready to explore our new home and all the things it has to offer. Yet, he doesn't quite know where to begin. Even before we moved, I could tell he was beginning a transition of some kind.
Anyway, after he told me what was going on with him, I nodded and said, "Okay." Then I got up and followed him. He walked over to the sofa, where he put on the television. We sat side by side for a while, watching. It was a show we'd seen before, but that was okay. He put his head on my shoulder.
As we watched, I looked around the room for something interesting that he hadn't explored in a while. I found something, went over and pulled it down. On the floor in front of the television, I began building. His face lit up and he joined me. We played like that for about an hour--watching and building and chatting about whatever came up. I could tell, when he walked away to to his own thing, he felt rejuvenated. He was looking for connection while he processed some restless feelings of uncertainty.
My son likes building and tinkering, so those are the kinds of things I look for when he seems restless. I don't ask him if he wants to do something. I just start doing something that I think he might enjoy (and that I enjoy). Most often he comes over and watches or joins in. I know he likes to be goofy, so sometimes I get goofy. I know he likes conversational games, so sometimes I start one. I know he likes to make food, so sometimes I'll offer to make something that he can join me in.
I've noticed with my son, those uncomfortable feelings of restlessness and uncertainty about what to do are eased by some company. Just me being fully present with him seems to help a lot. Sometimes a half an hour of doing something together is enough to spark an idea for something else he'd like to do. Sometimes he wants to spend longer periods of time together. I don't ask a lot of questions about what he'd like to do. I just spend time with him, sometimes doing things, sometimes just sitting with him. I aim to be a good friend to him in these moments, because I think that's what I would want most for myself.
Like Alex said, spend some time really getting to know your daughter. Find out what she enjoys about the shows she watches and the games she plays. Watch and play with her. Talk with her about her interests, and then add some things you think she might enjoy.
Try not to worry. I know that's hard. I'm a worrier, myself. But when we worry about another person, it becomes a burden for them on top of what they are already experiencing. Just be with her, as fully as you can. If she's telling you she's bored, she's inviting you into her experience. Join her. Learn about her. Share yourself with her too. You'll likely learn a lot about her (and yourself) in the process, and I'm confident it will be enriching and rewarding for you both.