I'd personally let the cold run its course. But, dependingon the age of your child and the severity of the cough, runny nose, I'd ask the pediatrician their advice.
My first instinct is that Benadryl wouldn't help as an antihistamine, but I'd also never think to use it that way. However, diphenhydramine, the active ingredient, is commonly used in other products such as sleep aids because of its sedative effect (check the active ingredients in Tylenol PM, Motrin PM, Advil PM, Unisom, etc).
I'd only do it under the blessing of my pediatrician, but my kids are young, so dosing is really important still.