Prescription cough syrups containing the narcotic hydrocodone can cause nausea as well as other side effects, such as constipation, dizziness, drowsiness, fatigue, vomiting, and psychological dependence. That's one reason that syrups containing hydrocodone—or the related ingredient codeine—should be used only for especially persistent and wrenching coughs, and only for adults.
Try milder measures first, such as humidifying the air to moisten dry airways, drinking hot liquids, and sucking on non-medicated lozenges. Most coughs actually stem from postnasal drip associated with the common cold. In those cases, using an over-the-counter antihistamine or nasal spray can help resolve the symptoms that cause dripping. Note that no cough product, either prescription or over-the-counter, is recommended for young children.