When I first read the question I told myself: OK, here's another of those stupid "natural"="good" "artificial"="bad" claims... however, it seems this is not the case.
There are indeed several studies on the matter, although they generally do not look only at food colouring, but also at other additives such as the preservative sodium benzoate or various sweeteners.
A randomised, double-blinded, placebo-controlled trial was published a few years ago on Lancet
Food additives and hyperactive behaviour in 3-year-old and 8/9-year-old children in the community: a randomised, double-blinded, placebo-controlled trial. - McCann, Lancet 2007
Jumping straight to the conclusions, they say that:
We have found an adverse effect of food additives on the hyperactive behaviour of 3-year-old and 8/9-year-old children. Although the use of artificial colouring in food manufacture might seem superfluous, the same cannot be said for sodium benzoate, which has an important preservative function. The implications of these results for the regulation of food additive use could be substantial.
So, what exactly did they do?
They recruited groups of 3 and 8/9 years old children from Southampton, UK. In particular they had 137 3 years old and 130 8/9 years old completing the study (the paper explains the recruitment in more detail). They then tested the effect (vs. placebo) of 2 mixes, called A and B, containing different amount of various food colouring (sunset yellow, carmoisine, tartrazine, ponceau 4R, quinoline yellow, allura red AC, plus sodium benzoate. 8/9-year-old children had 1.25 times bigger doses.
Now, I don't have kids myself, but reading this sentence I would say this is waaaaaaaaaaaaaay more than I would give my kid to eat... anyways
Doses for mixes A and B for 3-year-old children were roughly the same as the amount of food colouring in two 56-g bags of sweets. For 8/9-year-old children, the dose for mix A was equal to about two bags of sweets a day and for mix B about four bags of sweets a day.
The protocol they followed
After a week on their typical diet (week 0: baseline diet), the artificial colours to be used in the challenges and sodium benzoate were withdrawn from their diet for 6 weeks. Over this period when challenge with active or placebo drinks were given, additive withdrawal continued (week 1: withdrawal period but receiving placebo; weeks 2, 4, and 6: challenge with randomisation to two active periods and one placebo period; weeks 3 and 5: washout continuing on placebo). During this period, 3-year-old children received the challenge and washout-placebo drinks on a weekly basis and consumed mixed fruit juices (placebo or active) at home (300 mL/day for 3-year-old children, 625 mL/day for 8/9-year-old children), provided in identical sealed bottles. At the beginning of the study, children were assigned by the study administrator by a random-number generator to receive one of six possible sequences of placebo, active mix A, or active mix B challenges across weeks 2, 4, and 6.
All the various mixes were tested so that they could not be discriminated by look and/or taste.
Behaviour was then assessed with different methods (essentially questionnaire given to the parents, assessment of behaviour in classrooms and, for the older children, a visual test). I am not an expert on these matters, so I cannot really comment on these tests, but I assume they're quite standard ways to measure hyperactivity (and I'm sure Lancet recruits competent reviewers...)
I'll skip you all the stats, but the conclusion was that
Our complete case data has indicated that the effect sizes, in terms of the difference between the GHA under active mix and placebo challenges, were very similar for mix B in 3-year-old and 8/9-year-old children. For mix A, the effect for 3-year-old children was greater than for 8/9-year-old children. The effects for mix B were not significant for 3-year-old children because there was greater variability in the response to the active challenges than placebo in this age group. Thus, we recorded substantial individual differences in the response of children to the additives. For both age groups, no significant effect of social and demographic factors was seen on the initial level of GHA or in the moderation of the challenge effects.
They continue saying that:
The specific deleterious compounds in the mix cannot be determined for the present study and need to be examined in subsequent studies. The effect of artificial colours needs to be differentiated from the effects of preservatives in a 2×2 design. Further investigation would also need to establish whether the age-related difference seen in the present study can be replicated—ie, the effects of mix A being greater for 3-year-old children than for 8/9-year-old children. We examined the effects of additives on changes in behaviour during an extended period in a community-based, double-blinded, placebo-controlled food challenge. A weakness in this approach is the lack of control over when the challenges are ingested in relation to the timing of measures of hyperactivity. This study design also needs extensive resources to obtain multisource and multicontext measures of hyperactivity. We have completed a pilot study showing that changes in hyperactivity in response to food additives can be produced within about 1 h. Therefore, future studies could use more feasible acute double-blinded challenges undertaken in more controlled settings.
There is a big amount of literature on the matter.
I will only link to a couple of reviews, you can go through their references if you want more information.
Do artificial food colors promote hyperactivity in children with hyperactive syndromes? A meta-analysis of double-blind placebo-controlled trials.
Despite indications of publication bias and other limitations, this study is consistent with accumulating evidence that neurobehavioral toxicity may characterize a variety of widely distributed chemicals. Improvement in the identification of responders is required before strong clinical recommendations can be made.
Dietary sensitivities and ADHD symptoms: thirty-five years of research.
Artificial food colors (AFCs) have not been established as the main cause of attention-deficit hyperactivity disorder (ADHD), but accumulated evidence suggests that a subgroup shows significant symptom improvement when consuming an AFC-free diet and reacts with ADHD-type symptoms on challenge with AFCs. Of children with suspected sensitivities, 65% to 89% reacted when challenged with at least 100 mg of AFC. Oligoantigenic diet studies suggested that some children in addition to being sensitive to AFCs are also sensitive to common nonsalicylate foods (milk, chocolate, soy, eggs, wheat, corn, legumes) as well as salicylate-containing grapes, tomatoes, and orange. Some studies found "cosensitivity" to be more the rule than the exception. Recently, 2 large studies demonstrated behavioral sensitivity to AFCs and benzoate in children both with and without ADHD. A trial elimination diet is appropriate for children who have not responded satisfactorily to conventional treatment or whose parents wish to pursue a dietary investigation.
So... essentially there is not a crystal clear answer, but there may be effects of specific food colouring, possibly depending on the sensitivity and the age of the children.