So looking at Verizon's site there are some really good phones offered at zero cost, but it's really important to note that's a zero cost up front but you're still going to be paying for that 'free' phone, the cost being spread out over the monthly payments you have to make over whatever contract period. Verizon isn't going to be giving any phones away as some kind of social service, it's going to be making a lot of money on your account with them with that 'free' phone. It's a clever (a.k.a 'devious') way to pull in customers, a spin on the classic bait-n-switch tactic retail stores use.
It might be worth your time to pick one of the more viable options for your son, and look into how much you'd pay for that same model if you bought it as is somewhere else (i.e. Amazon). This is the hard part in that you need to guestimate the total cost of that phone over the time period of your entire contract with Verizon -- the messy part being the phone cost minus the cost of the cellular services used. Try looking into the 'bring your own phone' account cost as a comparison, again guestimating the price you'd pay for the phone plus the cost of the account usage. Lots of variables involved but the cellular companies add a lot of markup to those 'free' phones.
Also, keep in mind that carrier-locked phones are a) locked to that specific carrier so, just as an example, you find the Verizon coverage sucks you'll be paying big bucks as a 'early termination' fee to switch to a different, better carrier and b) typically loaded with a lot of carrier-specific bloatware. A carrier-unlocked phone allows you a lot more freedom (just be aware not only is carrier bloatware an issue but there's also manufacturer bloatware).
Anyway, there a number of good budget-class Android phones available these days. The news media and a lot of people fixate on just pricey flagship models but there's also reality and what the majority of consumers actually use:
Approximately how old is your son and what do you think are his expectations in a phone? Also, what about his friends and such? Peer pressure and phone status has become such an issue now.