Yeah, as Feather's suggested, for the price/size .. Sony's RX's seem to fit the bill well.

RX100's are compact.
RX10's are larger but with more zoom.

Quote Originally Posted by ameerat42 View Post

..... he began to lament how his recently acquired Nikon Coolpix S7000 just didn't cut it for image quality. We compared
fairly similar images of Portofino, his with the 16 Mpx Nikon Coolpix S7000, and mine with a 4.7Mpx Σ camera.

Quest.
After some moments of being quite gobsmacked he asked if I could help him "get a much better camera". .....
I think you need to disseminate this area of concern to determine exactly what would fit the bill properly.

That is, exactly what was it about the quality that he wasn't impressed with(compared to the Ʃ)?
Was it the detail rendering .. was it the colours .. etc.

Start with resolving that question which will help to narrow down the potential field a bit more.

OH! and a camera system that many folks forget about is the Samsung line.
While I don't have one myself(well my daughter has a Galaxy Camera) .. from what I've seen of them they look quite good.

If I were in the market for a compact capable camera ** ... from everything I've seen I'd be going with either a NX3300 or NX500(with a stronger lean towards the better looking NX500).
If you're interested: Have a look at DPR's new studio tool thingybob!
Load up the following cameras in the vacant fields. Canon G1x II : Panasonic GM1 : Sony Rx100 iv : Samsung NX500

To my eyes, from that selection of current cameras, the Samsung looks to have the best IQ in that studio scene.
(note that while that studio scene doesn't represent real life, it's the only means we currently have to compare apples to apples ... and not some random apples-blue cheese comparison
Play with high ISO levels etc, and the Samsung still comes out on top.

** note: I am toying with the idea of getting an NX500 as a replacement for my daughter's Galaxy Camera. Until something better comes up as an alternative, this is my current leaning.
She loves the Galaxy Cam, and doesn't want a too big camera(say like my totally idle D70s) and I also want her to have an all round better camera.
ps. while the galaxy is fun for her(and I have to admit it is fun to play with), she's getting annoyed with it's inabilities(focus, control etc). I hate it's lack of any half decent IQ and get annoyed with it's lack of most controls

So while some of those camera require a lens 'kit' (ie. are interchangeable lens cameras!) .. if you take the point of view that you only want one lens on it and leave it at that .. they need not be ILC cameras ..
Panasonic GM1 is about as small as you'll get in a larger than sub 1" sensor camera. Have a look at how tiny they are with a very basic small lens(12-32/3.5-5.6) It's basically pocketable!
Samsung NX500 + the kit 16-50/3.5-5.6 isn't too much bigger tho.

The hard part about Samsung cameras is finding a store that stocks them.
I had to search everywhere for one! .. found Camera House selling them for $799 which includes the kit lens + case + 1 spare battery.
The GM1 + kit lens is also at $799(but I'd have at least one spare battery on non OVF camera!!) so they're pretty comparable.

I suppose the selling point here(and I'm not trying to sell anything .. just making a point!) . is that even tho you said a capable compact camera, and not a compact ILC system .. don't look at it from that viewpoint. Look at it from the POV that it doesn't really matter as long as it's compact.
BUT! .. in the event that said relative actually enjoys taking photos with a renewed vigour due to the higher quality images now .. the option to update the lens only as opposed to updating the camera .. AGAIN! .. is now possible.
If there is no further interest in the hobby .. no big deal.

While FZ's and RX10's are great for their large zoom ranges, they are anything but compact!
While they are smaller and more convenient than a DSLR, they do have quite obvious limitations compared to DSLR type cameras. And if you check out size comparos, in reality they aren't all that different in 'relative' sizes anyhow!

If you want a fixed zoom lens type camera with as much as you can get in the one convenient package .. obviously the size will inflate as the zoominess of the lens increases .. I'd say going by all reviews that the Sony RX10 is the best way forward from there.