use printer ink, it will last longer and is more stable
Food coloring does work, but the only ones that didn't fade for me were the ones where I used a lot of it. For example, I used the McCormick neon green to get a nice dark green by using quite a few drops. I used the neon green because it initially looks kind of yellow, and this allowed me to keep adding it until I got the shade of green I was looking for. It's been several years with no discernible fading at all.
I also recently tried Sharpie purple highlighter (as recommended by someone else on this forum) to re-dye a couple of purple/yellow globes, and the initial results have been excellent. Too soon to tell if they will fade, but for now they look great!
I'd also recommend printer ink but be aware of the magenta. It often makes the lamp seem somewhat cloudy.