With the "Principles of the Box" in mind, then how does one actually go about the process? What follows is a descriptive process as opposed to a prescriptive process. In other words, it describes what I believe to be organically inherent steps in the pursuit of ideas rather than dictates a specific set of techniques to use while you're going through the steps.
The first phase is Input. As James Webb Young first articulated, this consists of both general and specific preparation. It consists of every fascinating conversation you've ever had, the books you've read, the movies you've seen, the people you've met, the travels you've taken, etc. It is pop culture and education. It is your life. It is also the information specific to the opportunity you're trying to address or problem you're trying to solve: market research, experiences with the product, service or issue in question, etc. Here is where you gather things in your box.
The second phase is Interpret. This is the conscious effort to identify what you have to work with and analyze (take things apart) and synthesize (put things back together) all the things you have in your box relevant to the problem at hand. It's typical to hear creative people talk about working on a problem to the point of exhaustion at this phase, and frustration is common. It's important to embrace that as a necessary part of the process.
The third phase is Incubate. This is where the process starts to sound like New Age psycho-babble, but time and time again some of history's most famous creators have found that the bridge between the frustration of consciously working a problem and the "aha!" moment is a period of incubation. This is about diverting the conscious mind from the problem. Taking walks, sleeping on it, turning to another project all seem to make a difference. From my own experience, I can cite at least one example where I literally solved a problem and developed a PR campaign in a dream as a result of this process.
The fourth phase is Inspiration. This is the only noun among verbs in the names of the phases because it is the one phase that creative people consistently report feeling "happened" to them rather than them having "caused" it to happen. This is the flip side of the Incubate coin because it is directly from Incubation that Inspiration arrives. It is the "Eureka!" moment when your subconscious delivers to your conscious mind the relationship that becomes the actual idea. It is the Holy Grail of the process and the least controllable. All you can do is embrace the process and be open to the moment.
The final phase is Integrate. This is where idea creation begins to transition to idea management. Here is where you test the idea against the constraints of the assignment or your vision. Is the idea executable? Affordable? Relevant to the target audience? Consistent with your artistic mission? On strategy? If the answer is "no" and the idea in question is genuinely a result of the process described here, the solution is rarely to go back to the drawing board. Rather it simply means that specific considerations have to be tweaked.