I'm a musician as well. Kind of a jack of all trades, master of none. I play Keyboard/Piano, guitar, bass, trumpet, saxophone, and "loosely" play the drums (by far my weakest instrument).
While I understand the desire to get an instrument economically, I personally don't trust the stuff at Target/WalMart when it comes to instruments. You can get a good quality, recognized brand acoustic guitar for a couple hundred bucks, give or take, at a reputable music store (local is better then big name chains). Fender Guitars comes racing to my mind, as that was my first acoustic and I don't think I paid over $200 if I recall correctly. There came a time when I had to sell it, and I still was able to get a decent buck for it. IMHO, that's not as likely with a WalMart/Target guitar. I don't even know who makes those. A good quality (even if inexpensive) reputable guitar is a far better investment then a chain store bought one. If it happens the instrument isn't welcomed, one like a Fender will be much easier to sell.
Musically, learning basics on a piano/keyboard would certainly be a good thing along the way too. If there is a musical interest, basic piano knowledge will assist in whatever instrument is chosen. I played piano before I played trumpet or saxophone. Reading the music for the brass was far easier then reading music for the piano, but the efforts in music theory with the piano more then paid off with the other instruments.
Another way to think of it is if your kid was into mechanical repair. As the kid was learning to work on engines, etc. you wouldn't buy them plastic tools from the toy section at a retail chain store. On the other hand, you likely wouldn't call the Snap-On guy to stop by in his rolling showroom to buy tools either. You'd likely go to a hardware store or home improvement store and buy good quality yet economical tools, such as Stanley, Husky, or maybe Craftsman. You gotta have good quality tools to do the job. No different with instruments. The "off brand" guitars I've seen are very poor quality, are often tough to tune, and sometimes don't hold a tune for long if they can be tuned. All of this can be discouraging for a budding musician.
Mind you this is merely my experience and personal preference/opinion.