The kind of rash that you have described sounds like hives. Hives are red, itchy welts or swellings on the skin that often come in clusters. They are also known as "urticaria."
Hives can appear anywhere on the body, cropping up either in one small area or covering large patches of skin. They most often occur as part of an allergic reaction.
In young children, a viral infection like a cold may trigger a bout of hives. Apart from this the most common triggers are Foods (commonly eggs, shellfish, nuts, berries, pollens and plants (nettles, poison ivy, poison oak, and so on), animal fur, insect bites or stings.
The swelling that you see is called angioedema; this will usually not be associated with itching.
It is similar to hives; the two differ only at the level in the skin, where the swelling occurs. Otherwise the same mechanism is responsible for the skin reactions in both the conditions.
Treatment is with anti-histamines like Benadryl, which must have already been prescribed. Comfort is the main goal of treatment, since most hive outbreaks are relatively minor and short-lived. To soothe itchy skin, try cold compresses, calamine lotion, a cool shower, or a tepid bath will help.