In this question, the OP asks about the problem of his wife's fingers being too short to cover the guitar's fretboard entirely. Similarly, my wife wants to start learning piano and her fingers are also short. I am just wondering if there is anything negative related to short fingers and playing the piano.
Naturally, someone with small fingers won't be able to make the larger reaches in some pieces. Some stretches aren't reachable even by people with average size hands. Not every piece is for everyone. In particular there are some Bach, Beethoven and Rachmaninoff pieces requiring stretches of an octave or more (though I couldn't tell you which ones offhand). Rachmaninoff could reach a 13th!
That said, your wife should have no problem starting out. As she advances there may be some pieces she has difficulty with; these can either be skipped or modified. This question may be helpful here: Well-known composers or piano pieces generally suitable for someone with small hands?
I personally know a wonderful pianist and composer with small hands, so it's certainly possible to do well in spite of it. As noted in the answers to the question above there are famous composers and players with small hands.
As for other potential problems, I guess that it may be easier to strain small hands. Stretching is important as well as taking appropriate breaks, and not trying to force more than one can handle.
My grandfather was a pianist with extremely large hands (I don't know what he could reach), but he always said the pianist with small hands were lucky. He felt his large hands were a disadvantage and he had to work a lot harder to get them to do what was needed. Reaching chords is just one small aspect of what is needed to play well.
Music is not like sports, where a 5-foot-tall basketball player or 7-foot-tall jockey will have difficulty making money at it.
With small hands and short fingers probably won't be asked to play the Rachmaninoff concerti with the Berlin Philharmonic, but let's face it, how many of us ARE asked to play the Rachminoff concerti with the Berlin Philharmonic?
For most kinds of music -- jazz, rock, pop, blues, country -- small hands don't matter. You can play the chords any way you want -- all spread out, or all bunched up within an octave. For these kinds of music, playing all the notes as written is not important; conveying the feeling is what's important.
Most classical pieces from Mozart onwards have octaves and will be difficult to play as written for someone who can't span a full octave. You can still play them recognizeably, leaving notes out. Most listeners who aren't themselves pianists will not even notice the difference. You won't win any competitions, but then if you're in music for the competitions, you probably should be in sports instead.
Trivia: Josef Hofmann had very small hands, but that didn't stop him from being one of the greatest pianists of all time.
I would not say there's anything negative associated with having short fingers or small hands. Alicia DeLarrocha, easily one of the greatest pianists of all time, was famous for having small hands.
She increased the flexibility of her hands through specially developed stretching exercises which I may have a copy of (email me, and I may be able to get a hold of them.) I don't know if they're available in print or not.
In any case, your wife can always just arpeggiate chords that are too big to play. Lots of pianists have to do it, you just have to learn to be clever about hiding the fact that this is happening. Break chords up into groups instead of just "rolling" them. Also, experiment with arpeggiating before and on the beat. Keep in mind that arpeggiating on the beat will cause a tempo distortion.
If she can get up to an octave, she should have no problems playing at all. (I can reach a 12th, and I still have to arpeggiate chords from time to time :).)
With less than an octave span, you are going to have problems for quite a bit of literature.
There are a few keyboard types friendlier to smaller hand spans, but you won't find most of them, with the exception of Jankó keyboards, on a piano or grand piano. But that's not really an option when you don't play in a "bring your own piano" setting.
Note that the largest key width (and the only more or less standardized one) tends to be the grand piano, and that's what's used in concerts. Upright pianos might be slightly smaller, but when you are going for concert play, you have to scale a grand piano.
Smaller keyboards tend to be on harpsichords. They don't have the dynamic possibilities of a piano but still have a percussive character. Moving over to aerophones, you get harmoniums. You lose the percussive character but have a better chance of actually dragging your own instrument around given a reasonably sized convertible car and willing helpers.
Of course, the next smaller size is an accordion. Chromatic button(!) accordions with "converter mechanics" like a bayan can actually be used for executing an astonishingly large range of classical piano music while making do with much smaller hand spans. You can also "easily" carry them to performances as they are "compact" and don't tend to weigh more than something like 16kg. Slightly lighter than a typical electronic stage piano but less bulky.
However, the lack of per-note dynamics and percussive character make a lot of difference regarding the resulting character. There are a few recordings of the "Wohltemperiertes Klavier" and "Goldbergvariationen" by Bach around that are worth checking out and work out pretty well. Mozart tends to work out pretty nicely, too, and I'm rather impressed with how Grieg turns out (no idea whether anybody except myself is trying him, and as opposed to Mozart or, say, Strauss or Waldteufel or Scott Joplin, you definitely need converter type accordion for Grieg as the standard accordion bass chords are entirely useless for him). However, I should think that stuff like Rachmaninoff or Chopin make for quite a worse match.
One reason is that the left hand is separated from the right hand and the left hand buttons are less suited for virtuoso play than the right hand ones.
At any rate: the break-even point for seriously thinking about picking a different instrument altogether is likely whether or not you are able to stretch to an octave on a grand piano. Anything less will severely impact your choice of music and literature.
It is generally not much of a problem if you can reach an octave with the pinkie and thumb.
You can compensate somewhat for small hands and lack of reach with fingering changes, such has using the other hand for one of the notes, and occasionally leaving out a note.
I have discovered that Franz Liszt had significantly larger hands than I do!