You will need a plastic cup for each family member and a paint pen. I’ve tried permanent markers (like Sharpie) in the past, but that washes off after a time or two. Paint pens work better—look for an oil based, permanent, fine point paint pen.
Some other ideas…
Colored Cups—if each child in your family has a designated color than it’s easy to get a different colored cup for each child. (I wanted to avoid the arguing over different colored cups so ours are all the same color!)
Kid Decorated Cups—have each child decorate their own cup with their name and a simple pattern or drawing. My mom had us do this when we were younger—I drew a bunny on mine! We mostly used our name cups for road trips and camping.
Water Bottles—this idea also works great for refillable water bottles! We have a set of nice ones we use for road trips.