Violin, in spite of its reputation, doesn't have to sound bad in the early stages. You need a good teacher, an enthusiastic student, and some involvement from the parents. There will be a few squawks and struggles with intonation, but the "strangled cat" image is not fair!
It is difficult to learn how to hold the violin, but once you can play a little, you have so many options as far as types of music and groups you can play in. It also has such a beautiful and versatile sound once you can play really well. The flute sounds lovely, but has less variety and colour- always sounds like a flute, which can start to get tiresome. (This is why you will always see classical chamber music concerts, recitals, and concertos which feature violin, but a flute recital is more rare.)
One thing to remember is that an orchestra only needs 2-4 flutes, and about 30 violins, so there is always room for another violinist in school or county orchestras. Looking ahead, if your child ends up loving classical music, the repertoire for violin is just hugely more interesting and fulfilling; there is no comparison.
Having said that, I think the flute is easier to start (if she is big enough), looks and sounds lovely, and is less of a commitment in that you can progress more quickly with less effort. It is easy to carry, less fragile, easy to tune, and something a child can do with less involvement from parents. I think for a child who has busy parents, lots of interests, and no particular passion for the violin, flute is a good choice.