The few historical paintings I've seen with children and rosaries in them are either paintings that (anachronistically of course!) show the Virgin Mary with the Infant Jesus on her lap playing with a rosary, or else like this one they show the child with something that basically looks just like an adult's rosary. (This is the Dauphin of France, painted in 1494 by the Master of Moulins.)
While there is no formal age in the Catholic church where a child is given a rosary, it's fairly common to see "children's rosaries" advertised for sale. About half of these I would actually say are more "in honor of" the child than actually for him or her, since they are generally flowery, sentimental items, perhaps given to commemorate the child's baptism. Some of them are fragile, heavily decorated or otherwise clearly more the sort of thing you'd hang on the child's wall rather than give it to her to play with or (horrors!) chew on!
Most of the remainder, sometimes referred to as "toddler rosaries", are made from large wooden beads with a plain cross, and clearly are intended precisely as chewable, playable toys, although of course families do often encourage children to join them in praying the rosary as soon as they're able to understand it. I've seen these in bright primary colors, and more recently, also in uncolored and pastel versions.
The earliest formal occasion where a child may be given a "serious" rosary is often around five to seven. In current practice, this is old enough for the child to at least learn and repeat the prayers along with others, and around age seven the child is usually introduced to the formal sacraments of Reconciliation (formerly known as "Confession") and Communion.
Children this age are often simply given the same sort of rosary one would give an adult. You also see the decorative, "dressy" sort of rosary (perhaps white, sparkly, or embellished), especially as a First Communion gift. I've also seen the so-called miniature rosaries referred to as "children's rosaries," perhaps with the thought that they are better suited to a child's smaller hands. (I suspect the real reason, though, is that they're simply cute!)
One eBay auction says, "The devotion of the rosary has a special connection with children; most of the Marian apparitions are first experienced by children. In many of these apparitions, Mary encourages the children who see her to pray the rosary, and she often appears with rosary beads on her person. A child, perhaps, sees better than we grownups do, and can truly appreciate what is holy."