Welcome to our community!
I assume that you two are not married, right? That would be the most convenient way to solve the issue since being married to a German grants you the right to stay here.
If not, having a child with a German woman will not grant you the right to stay automatically.
There is a way to get a residence permit if you decide that both of you will take care of the child. It is called "Aufenthaltserlaubnis zur Personensorge". The problem is that this requires that you are able to earn a living without the need of getting benefits. That is not possible if you only have your blue passport because you are not allowed to work with it here. You need a work visa in this case.
Furthermore you need a valid passport to prove your identity (maybe Italian refugee status and/or blue passport is enough) and a permission to be here in Germany (for example said visa) to go through the procedure if your stay lasts longer than three months (your blues passport only allows stays in other EU countries up to a maximum length of three months and - as stated above - you are not allowed to work).
If you fulfill the criteria listed above then you can apply for this type of residence permit at your competent Ausländerbehörde.
You can declare that both of you want to take care of the child at the same time as you do the "Vaterschaftsanerkennung" (acknowledgement of paternity) at Jugendamt. You have to tell them that both of you want to get the so called "Sorgerecht" (child custody). If you already did the acknowledgement of paternity you can still go to Jugendamt and tell them.
I will link @Marcel and @Steven to see if they have an idea how to handle your case best.