First, as with any beginning fly angler, you're probably thinking that you don't want to spend the money on a high end rod as their fist tool of the trade. That makes sense, especially if you're not sure if your child will stick with the sport, but you certainly don't want to give them junk to begin fly fishing with either. Of course, if you did end up getting them a mid to high level rod & they decided that fly fishing wasn't their cup of tea, you'd end up with a nice extra rod for yourself. So there's logic in both approaches here & rods like my Prudential Series seem to bridge that gap nicely.
As for the rod to get them, I'd suggest that you pick it out with this one rule in mind: make sure that it would be a rod you would fish with. That's worth repeating - don't get them a rod that you would refuse to use yourself. It doesn't have to be as smooth in action or as nice looking as yours, but it should be a rod good enough that you could get through a full day of fishing with. A poor fly rod will make the learning of the sport that much harder & we want them to enjoy this.
I see on the market some rods that claim to be youth fly rods. Some of these are made up in 'fun' colors. More often than not they are atrocious as fishing tools. I couldn't recommend you wasting your money on them as they are designed to catch you - not any fish. Some companies are offering real rods marketed towards younger anglers that actually look like real fly rods. These might be better than the others, but I would urge you to look them over before you buy.
I was able to use my first fly rod well into adulthood without embarrassment or disappointment. It didn't cost a fortune back when it was new & it worked well then & still does today. My folks knew what to look for to make sure that the rod wouldn't be another obstacle in learning the sport. That's what you are aiming to do with your kid, right? So just make sure that you take your time in choosing their first fly rod. Years from now they will be glad you did.