I've struggled with this problem since the iPhone 5 first came out, and if I can't find a solution I'll have to (reluctantly) change phones. I'm a decent amateur photographer so I don't expect miracles from the iPhone 5, expecially indoors, but I expect a lot more quality than I'm getting. Outdoors, the iPhone 5 is great...no noise at all. But indoors, even on a steadying surface, the pics come out horrible. Here's a link to a typical indoor picture in a very well-lighted room without flash. I've had a few pictures turn out better than this, but most are this terrible.
FWIW, I used to have an iPhone 4S and it took very good indoor pictures.
It doesn't look like a well lit room in the picture but, of course, I wasn't there. It looks to me like an iPhone picture taken in a dimly lit room. Steadying the camera isn't going to make any real difference in noise. It will cut down on blur in a possibly longer exposure time.
If you think that there is something wrong with the camera, that it's not performing to spec, you should make an appointment at the Genius Bar at your local Apple Store. Your phone should be under warranty.
Best of luck!
Meg, if you'd have been there you would have seen how much light there was in the room. It wasn't like being outside, but it was certainly not dim. BTW, I understand that steadying the camera has nothing to do with noise, I simply added that for clarification.
What I didn't add to my original post was that others at our table at this wedding were taking camera pictures with different phones, and though I don't have access to them to provide a link, the ones taken with a Samsung Galaxy S4 were nearly as good as some of the pics I took with my Canon. So there is a difference in phone cameras, and for a test last night I took some pictures with my wife's iPhone 4S and compared them to my iPhone 5 and the ones from the 4S were definitely less noisy.
The iPhone 5 was released on 21 Sept 2012 so yours is still under the original warranty.
I don't have the problem you have (nor, I think do most people) so it's not a problem with all iPhone 5 cameras. But, it is, of course, entirely up to you as to whether or not you investigate the hardware problem possiblitly. If you just need a reason to buy the new phone, I won't tell!
Best of luck.