Based on your comments, it would be a mistake for you to buy a multi-lens camera of any kind, no matter the price. None of those camera will fit in a pocket or be as convenient as your phone. If you don't understand the specs of the various models others have suggested, very they are irrelevant to your needs.
The V1 and NEX cameras are nice, and some sell relatively cheap right now, but any of them will set you back by at least $400, and you'd likely never use many of their features.
You want, as you stated, a point and shoot camera that costs little and takes better pictures than other stuff in its league.
If $600 for a Sony RX100 is too much for your budget, then I'd suggest a Panasonic LX7, whose Amazon price is currently around $320, but which may fall to $275 or less on Black Friday. The LX7 auto mode is pretty good and it truly is a lot of bang for the buck.
The Canon S120, Olympus XZ-2, and Nikon 1 J3 are also decent cameras, but cost more.
By the way, I don't recommend buying any camera as a gift for anyone. Ordinary people don't need or use dedicated cameras any more. Enthusiasts, on the other hand, are better off picking their own devices.
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