If she seeks asylum in Germany she will go through the Dublin procedure. As she has a residency status in Italy, she would be deported to Italy where she then can have an asylum case heard there.
Does she have permanent residency in Italy? It is called ‘soggiornante di lungo periodo’. If she has this, then she can come to Germany and **** for a job here. If she finds work, she can apply for a special residence permit for people who have permanent residency in another EU country.
If she comes to Germany with her child, then it is possible under certain conditions for her to receive a residence permit as the parent of an EU citizen child. If the father also comes to Germany and is working here, then this is relatively straightforward. If she came alone with the child, it is more complicated and she should inform herself fully before undertaking such a plan.