Get a book light! A simple off-the-shelf book light produces just enough light so you can see what you're doing, but still keeps the room dark so your child can sleep through the process.
You can clip the book light to the table, to a changing pad cover, a bed sheet, or just about anything. The flexible part also helps in positioning the light just right.
Newer book lights use LED bulbs which are safer and produce no heat.
Great idea, although once the baby gets grabby, the light will be the prime target. Here's where a head-mounted flashlight might come in handy. Seriously! We own one and I can't tell you how many times we use it!
Does your kid sleep through diaper changes? How do you manage it?