I would think the piano would be easier for her to play. A guitar requires hands, wrists and finger to be coordinated in order to play the correct notes and chords. A wrist turned at the wrong angle will cause pain in the wrist. Beginning fretting of the notes and chord can be painful as calluses are formed. Improper posture can lead to back discomfort. If you go with a guitar, pick a teacher that is knowledgeable with children and proper technique.
There are smaller sized guitars for children. Different sizes include 1/4, 1/2, 3/4, 4/4. 1/4 is a good sized guitar for a very young child. When selecting a guitar, it is best to find someone who is knowledgeable and can help you pick out the right guitar. You would want your child to come with you are try out the guitar herself. Let her hold it, strum it, etc. She is going to be the one playing, so she needs to be comfortable.
I would advise against a ukulele or a guitalele. Uke's are a different instrument. They are smaller, but they only have 4 strings as opposed to 6. The chord fingering is different on the instruments. If she learns to play something on the uke, it is not going to transfer over to a guitar as far as where her fingers fall. She would then need to learn where to place her fingers on the guitar. A guitalele is a mix of a guitar and a ukulele. However it is tuned different, it is like playing a guitar with a capo. The chord shapes are also different.
It is not a bad idea to begin her in piano. I strong foundation in the piano will only help her play the guitar if she chooses later.
Also to add. Try to avoid excessively cheap guitars. They are often terrible sounding. They also tend to have higher action (which is how high the strings are raised off the neck of the guitar). The higher the action the harder it is to play. You need to press down harder to get a full sound, which causes more discomfort. Some of the cheaper guitars are First Act, guitars found in toy stores, or Target, etc. Buy from a music store.
EDIT:
While a guitalele can be tuned the same as a guitar, the sound is muddled. It is just not the same. If one is going to go to all the trouble of purchasing a completely different instrument, than they might as well simply get the actual instrument scaled down to 1/4 size. To give a child a step down (the instrument itself is not a step down, but it is being presented as such), is not going to instill confidence. She wants to play a guitar, not a guitalele.
If there were little options for getting an appropriate sized guitar, then I would agree that a guitalele would be a good choice. In my opinion buying a quality 1/4 sized guitar is the best option for a child that wants to play a guitar (not a guitalele or a uke). There are many options that will make playing an actual guitar easier for a child. As I mentioned above, getting a quality instrument, allowing the child to test out the instrument themselves, taking the guitar to a luthier for a set up (as most guitars are not set up perfectly right out of the box), choosing a guitar with low action, obtaining the right teacher, and picking the proper size (1/4 guitars are made for children that are bout 4-6). All these things will make the guitar easier to play, which is instill confidence in playing.
Not all children are guitar prodigies as you write you son as. A child that is able to pick up a uke at 18 months, progress to playing by 4, is probably going to have a lot less difficulty in transferring from one instrument to another.
I would suggest the mother speak with the guitar teacher that she chooses for her child. If the teacher is willing to train the child in a different instrument (and waste time), then it is the right choice. However, it is probably unlikely that option will be suggested.