There was some research in the news a while back which suggested that babies and young children were more susceptible to the "radiation" (or whatever it is) that mobile phones emit because their skulls are thinner. Not proven yet that this in turn causes problems (although there is some suggestion of tumours in the area around the ear being more common in regular mobile users) and not sure how this could apply if the phone was switched off.
eg.
"The findings show that parents are ignoring official health warnings about the hypothetical risks of mobile phone radiation on developing brains. Psychologists are also concerned that overprotective parents are "mollycoddling" young children and denying them the freedom needed to develop into mature teenagers." (telegraph 20.4.05)
and
"In the meantime, Sir William recommends that a precautionary approach be taken, especially with regard to children. The report states that children may be more vulnerable because of their developing nervous system, the greater absorption of energy in tissues of the head, and a longer lifetime of exposure. The Group has recommended that the widespread use of mobile phones by children for non-essential calls should be discouraged. Sir William also recommends that an information leaflet be made widely available to the public, setting out the report's advice." (dept.of Health 11.5.2000)
and
"Of particular concern to the NRPB (National Radiological Protection Board) is a report published by researchers at Sweden?s Karolinska Institute in October, which showed that people who had used mobile phones for 10 years or more had an increased risk of developing acoustic neuroma, a benign tumour regarded as serious because of its position next to brain tissue." (Sunday Times, 9.1.05)
You can tell I haven't got enough to do with my time can't you.