A body of research has elaborated and refined the effect of environmental cues on behavior. We're not aware of many of the things that influence our decisions, interests, and actions. Our minds are processing information all of the time, often below the level of awareness. We can see the brain lighting up in brain imagery scans, but when the researchers ask the subject what just happened, he will say, "Nothing." We also know that if we present some cue too quickly for a person to notice, it can still influence what she does, or a decision she makes, even though she could not tell you what the cue was. For example, a faint smell of cleaning liquids makes people straighten up their desks a little bit more than usual. When asked why they did it, they do not report on the smell, or they give a really good reason that has nothing to do with the smell.
In terms of motivating a child, it's useful to leave some novel and engaging things around the house where your child can come upon them in the course of daily life: books, a magazine or two, or a keyboard plugged in and ready to play. They should be things that don't require a lot of effort to fool around with (as opposed to, say, clay or paints) and can be casually picked up for a low-stakes initial look. Such casual opportunities to take an interest can exert a significant effect, but it's important to let your child connect with these engaging things on her own. Think of her just taking a brief peek at a magazine, a light connection. If you burst into the room at that point and exclaim, "Did you see the pictures of Wyoming? Weren't they gorgeous? I've booked tickets for us to go there this weekend," you're very likely to kill the budding interest. Instead, let the child find her own way to engagement; use the power of modeling to show her how: Sit down at the keyboard yourself from time to time, tinker and play a bit, give your child a behavior to imitate. (And, as we said before, you'll also have to draw the line on TV, video games, and other such distractions.)
Peers. Making your child's peers welcome in your house is good policy. For many children and families, the home is a place to sleep and to grab meals and not much more. But having the home open to peers makes it more central to family life and gives it a chance to exert a stronger continuing influence on a child. Also, as a child reaches adolescence, it's normal for him to draw a line separating you from his peers and whatever they approve of. Bringing more of your child's world into your home is a way to undraw that line and to monitor and protect him without babying him. It lowers his chances of going off course as peer influences pull at him and foster risky behavior.
Also, when you're choosing an outside-the-house activity, let the child select a friend to go with you. This might be one of the regular activities we mentioned under the heading of modeling—a concert, a movie, or some competence-building activity during which interest, knowledge, and skill can accumulate. Take your child, let her invite a friend, and let them both see that you enjoy the activity yourself. Have a little lunch or snack; make an excursion of it. As your child grows toward adolescence and her peers become more important to her, you can recognize and integrate that. At some point, the need for you to model an interest will fade away as the kids develop their own strong preferences and start picking the concerts—and you may have to suffer through some teenybopper extravaganzas that you'll have to chalk up to parental sacrifice. But the larger principle remains the same. By making a routine of accepting and integrating peers in family activities, you've taken steps to keep your child close to you and build your relationship while also letting her choose to be with her friends, and you've done it in a way that doesn't set up a conflict between the attraction of peers and your desire to cultivate your child's interests and motivation.
Here's a list of further resources on children and motivation.