I'm a bit confused as to whether the question is about variations or composing in general.
I think of, and to be honest, pride myself on that I am an aesthete.
I always have a concept or theme in mind of what the music, to me anyway, is meant to represent and the general vibe it's meant to have. A lot of times, I can basically hear it and see it in my mind before I begin to actually work on it. Often, for months before I work on it and it can take several drafts before I really "tap into" whatever I'm imagining. I don't know how to describe it, I assume other people feel it, but for my entire life whenever I'd hear music, I always drifted off into some reverie about what I was hearing — especially in instrumental music. I'd hear new age tunes by Yanni, Chris Spheeris or other Rompler-driven pieces from back then and I'd always see these different colors and images that went with it to me. Visions of soaring over castles and oceans, a romantic desert adventure, sunsets etc. Still do.
So for me, composing or songwriting has always been that I have this imagery floating around in my mind and I'm hearing some piece that goes with it and the whole "process" is trying to make that real.
Sometimes, the starting point is a painting or photograph. I'm sure some psychologist or whatever would call me "autistic" or people would consider me some sort of weirdo if they knew, but like I said I'm aesthete; I become obsessive over things that I think are beautiful. Even now, in some windows on my computer's taskbar, I have some paintings and stuff open and they're there for no other reason than to stare at them periodically, whilst listening to music or whatever. I just get lost in them; they are snapshot in time but after becoming so familiar with and enchanted by them, in my mind, they begin to move. I can imagine what comes next or what happened before. What the "story" of the picture is.
I read the Witcher saga a few years ago, and I paid good money to import the European (specifically British) editions of the books for no other reason than because I love the art on the covers so much that this Spanish artist made. I especially love this painting of Yennefer.
For me, this painting is just so striking. I've spent so much time gawking at it. I love that it's dark and haunting, but there's something romantic, warm and inviting about it either way that sends chills up my spine. Looking at it, I hear this music that has a sorrowful, but yet seductive violin and a woman's voice in the distance, accompanied by string instruments like lutes and classical guitars.
I'm still working on that tune, three years later and I haven't quite tapped into it yet, but one day...
I actually don't even see a point in composing music without first having that sort of inspiration, to be perfectly honest.
To me, if the music isn't as aesthetically-pleasing as the images in my mind, then there's just no purpose. Beauty inspires beauty, essentially.
The natural beauty of the world and fantasy provides infinite possibilities to inspire a meager amount of notes to reach the sensory heights; the sense of wonder, awe, and lifting of spirits that one gets when gazing upon a great baroque Palace, soaring over the alps or sitting at "Pretty Place" chapel in the early morning when the clouds are low in the valley below, and it fills you with the sense that this is a close as it gets to heaven on earth.
So for me, that's the method to starting a new piece. Something great, visually or sensation enters my mind, and then I use my knowledge of the craft to bring it to life. And without the craft, you really can't achieve this.
I believe it was Brahms who said that "inspiration without craft is but a reed shaken in the wind".