I did just give this a shot very recently and here's how it went:
Before anyone starts trolling: I'm not complaining, I knew I was rolling the dice when I started down this path. I'm simply sharing my experience.
1. Did some searching around in Amazon and when you go looking for a refurbished iPhone on Amazon you have the following choices:
-Renewed
-Renewed Premium
both of these are available through various sellers (including Amazon).
I thought it prudent to go with a Renewed Premium (shipped from + sold by Amazon).
2. Did some googling around for reviews of Amazon "Renewed Premium" iphones and all of the videos made it seem indistinguishable from Apple Certified Refurbished. In the videos I watched, the reviewer has a damn-near mint looking phone and is very well pleased with the deal. Note: these phones come in a box marked Amazon Renewed (to their credit amazon makes this explicitly clear). Unboxing is nowhere near as impressive as with Apple. And even though I was buying this as a gift... I was okay with that.... I have boxes for old iPhones and was going to use one of those.
NOTE: amazon also offers free returns for a full year after purchase (assuming returned in same condition).
NOTE: both sides were 100% in tact: absolutely no broken glass. It's simply the chassis that was in poor condition. Also, I turned it on once and it booted into the "welcome" or "hello" screen or whatever just fine and the screen had no obvious problems. That's as far as I took the inspection, though, because the phone had already failed my inspection.
3. Pulled the plug and bought bought a recent model. I got it out of the box and for a phone that was being marketed as Renewed Premium is was just too obviously damaged.
4. I immediately requested a refund. When you do so, the return gets a time line that lists all of the significant events: when amazon picked it up (via UPS), when they received it, when they issue the refund and when the refund is credited to your account. The original timeline is exceedingly long: it CAN take up to 30 days for this whole process to play out (maybe longer, I can't remember what the timeline first said when I checked it). Mine did not: mine only took 8 days start to finish.
5. Most events in the return process went much faster than the original timeline would lead you to believe. Everything happens very quickly up through the point where they tell you the refund was "Issued". A day or two between steps at this point. As you can probably guess, though, there were FIVE days between the time where the refund was "Issued" and when the refund was "Credited". And weirdly enough, when it actually hit my bank account (which I was watching daily) the refund was BACK DATED to the date the refund was issued. Weird. Again, not complaining, just sharing.
My conclusion: I would certainly not ever consider anything amazon renewed (premium or otherwise) as a gift unless the recipient is absolutely cool with this (has specifically requested an iPhone of any particular condition). Also, I'm put off by Amazon's Refurb process. If the phone I got was indeed a Renewed Premium, I shudder to think what condition a "regular" Renewed phone must be in.
I think amazon needs to provides picture samples that relate the actual condition of their phones to their "grading" method (which they do try to explain on their website). I might try again for a phone for myself because I kinda don't care if it has scratches, etc.
Final Note: Youtube reviews of Amazon Renewed created an expectation that my actual experience of an Amazon Renewed Phone did not match. Also, the savings are considerable but don't go thinking you're getting a great deal: this phone was not a new phone, it certainly was nowhere near Apple Certified Refurb quality and considering the actual quality of the phone I received, I think the price is still a bit on the high side.
