The short answer
Yes, often, and sometimes beautifully. Arabians have a long history as family horses, and many form an intense, loyal bond with a young rider that bigger, more phlegmatic breeds never quite match. But this is the pairing where the individual horse matters most of all the breed-and-children questions, because the Arabian's defining trait, sensitivity, can be either the reason the match sings or the reason it fails.
Sensitivity with small riders
An Arabian notices everything a child does: the light weight, the soft hands, the sudden shriek of delight, the flapping anorak, the pony-mad friends crowding the gate. A settled Arabian takes the good with the bad and answers the child's lightness with real generosity, which is why young riders who click with one progress so fast. A sharper individual reads the noise and unpredictability as a reason to be on his toes all the time, and a tense Arabian under a small child is a combination to avoid. Judge the horse in front of you, in a noisy environment, with children around him, not in a quiet one-to-one viewing.
The size benefit
What Arabians have over almost every other light breed for children is stature. Most stand fourteen to fifteen hands, noticeably less horse than a sixteen-hand warmblood or a big quarter horse, and that changes daily life: an older child can reach to groom the whole horse, lift the saddle on, and lead him without the rope being above her shoulder. Their build is also narrow compared with cobby types, which suits shorter legs. For a child growing out of ponies, a right-sized Arabian can feel like a real horse she can still actually manage, rather than a monument she is perched on.
Picking the individual
Look for the Arabian who has already done the job: lesson work, family hacking, years of being handled by whoever was around. Ask what he has done with children specifically, how he behaves at shows or busy yards, and whether he has ever been ridden by a nervous or very light rider. Watch him being caught and tacked up by your own child before you buy, and take more than one trial ride in more than one place. The famous Arabian loyalty cuts the right way here: a horse who decides your child is his person will look after her with real devotion.
Growing up together
Because Arabians commonly stay sound and rideable into their twenties, the right one can carry a child from first canters through teenage competition without being outgrown in ability, only sometimes in height. That long partnership works best when the whole household handles him consistently, using the same cues and respecting the quirks written down for him. Through Family Sharing in KinsLe, the owner can give chosen family members access to this sensitive horse's profile, sharing that profile with them for as long as the household wants them included.