The next step
These cameras should please anyone you know who's getting frustrated with their phone.
Panasonic Lumix FZ1000
Miss having a zoom lens and you want better photo and video quality? The FZ1000 is a fantastic option.
Fujifilm XQ2
It doesn't have all the bells and whistles of other enthusiast compacts, but the XQ2 also isn't as expensive. If you want better image quality than your phone, but still want a simple point-and-shoot experience, this is a safe bet for the money.
DxO One
Know a photographer who insists that all they need is their iPhone, despite the fact that they're not happy with a lot of the photos that come out of it? Get them the DxO One, a camera that attaches to any modern iPhone or iPad (the one with the Lightning connector) and let them take their photography up a notch.
Olympus PEN E-PL7
Retro-attractive, compact and full featured but friendly, the E-PL7 makes a solid choice as a gift for a first-timer who wants to move straight to an interchangeable-lens model.
Sony Cyber-shot RX100
It's a bit old, but this veteran compact camera was top-of-the-line when it came out and at its current price makes a great gift for someone who wants to graduate from their phone's camera.
Samsung NX500
When the phone isn't enough because it can't capture kids in action or decent bar and restaurant shots, the interchangeable-lens NX500 is an excellent gift for an upgrader who wants to jump past basic and into all-around-powerful without sacrificing the phone-like operation.