Will letting my baby sleep on his side cause a stiff neck?

No; not a good idea. Letting your baby sleep on his side shouldn't cause a stiff neck, but this position is not recommended for sleep because of the risk of sids. Babies should sleep on their backs; the incidence of sids is lower than if babies sleep on their sides.
Back is better. It is recommended that babies be put on their backs to sleep because it may decrease the risk of sids (sudden infant death syndrome.) allowing infants to sleep on their backs or sides is not recommended.