The process hasn't changed all that much in the last 75 years or so.
The great copywriter James Webb Young outlined his process in his 1940 book, "A Technique for Producing Ideas":
Gather the raw material. There are two kinds: specific and general. The specific is everything about the product or service and its consumers. Take your time. Immerse yourself in it. The general is your larger knowledge of life and the world.
Digest the material. "What you do is to take the different bits of material which you have gathered and feel them all over, as it were, with the tentacles of the mind."
Do something else. Take a walk. Nap. Go for a bike ride. It's harder than it sounds. This is always where the panic sets in.
Aha. The idea comes. Hello, there. It may be shy.*
Work and rework the idea in "the cold, gray dawn of the morning after."
These days, you'd add a step 6 (like Copyhackers' Joanna Wiebe recommends) for testing:
- Test copy via A/B testing (Optimizely) or just running a bunch of Facebook ads and measuring results. I'm doing that right now with a barbershop client.
- You can do usability testing via a service like Usertesting.com or you can take a bunch of hard-copy screenshots into a bar and tell people you'll buy them a drink if they spend a few minutes telling you what they think the websites are all about.
Otherwise it's about the same.
*I had an idea that came to me once on a bike ride about a program for habit change. The idea seemed so silly that I tried to ignore it for a while. Then I gave up and wrote a mini-book about it. Check out what it did for my business partner's pullup max.