Once upon a time, downtown Mesa was drenched in the glow of neon lights. Businesses like motels, diners and cocktail lounges buzzed with vibrant color as their illuminated signs beckoned the public.

If you’ve ever looked at a flickering neon sign and thought, I bet I could make one of those, be advised: You almost certainly could not. At once a science and an art, neon demands an artisan’s ...