"A" in this situation is just one way you can refer to a person that is in some particular state. In other words, you can attach an adjective to that person and use "a" to refer to him/her. This is not, therefore, something that refers explicitly to family, and in fact in almost every other situation it would be wrong to say "a Dudley." I imagine that the reason that it's ok to use "a" in this context is because it refers to one of potentially many "different" Dudleys, so you can have a happy Dudley, a sad Dudley, a sleeping Dudley, or even a screaming Dudley. Using "a" like this also lets you keep the word order how you want it. For example, say the author really wanted to keep the phrase "screaming Dudley." There is no other way to say it; the sentence structure would have to be changed to something like "as she wrestled Dudley, who was screaming, into his high chair." The original just sounds better in this situation.