Well, first of all, it's not the fathers any more than it is the mothers, and most often the actual pressure comes from the extended family.
Secondly, investing in women's shelters and other such and educating young women about that these places exist and why they exist, encouraging young people to find an identity that isn't directly connected to their family, providing ways (ie, money) so that women aren't in a position of dependence to their family (which is what in many situations prevents abused people from leaving). Generally feminist things, but there's also the legal side - making sure witness protection services actually works and that restraining orders actually keep people away, making sure there is a legal response when people feel threatened, and making sure that crimes against women - and not just the target group for honour crimes, but all women - are actually being taken serious by the police force.
Much the same would go for young men - It is very often the brothers that get settled with the task of "correcting" their sisters, or at the very least taking the legal blame for the crime, regardless of whether they think there has been anything wrong or dishonourable done. They also need to find an identity disconnected from the family, so that they have a stronger ability to resist when the whole mountain of family disapproval comes bearing down on them; and likewise, investing in their ability to create lives of their own (it's much easier to fight back if you don't live in the same house or eat food they paid for), especially in Italy, where I believe young people already have economical problems with leaving the nest, if I remember correctly Italy is the European country where people stay with their parents the longest. I figure you might want to engage them on the ethical/moral/philosophical level as well but that pretty much go under "find your own identity" in my book.
Finally, training people of different occupations to identify signs might be a good idea (cops in particular wouldn't be able to see much, though, unless Italy has police that interact with the same people and neighbourhoods on a regular basis; they just wouldn't see people often enough to have a chance to identify anything), but these things, like all kinds of abuse, generally aren't very obvious. It isn't a thing you can just ban away, though, it's not as easy as that (and certainly no by banning sharia, they're only connected by how they're both patriarchal systems - in some places in the middle East you'll find the Muslim Church Mosque being the main, or even only, fighter against the honour tradition) or we would have made such laws to ban oppressive sects and the like a long time ago. You can ban the actual harmful and violent actions, but you can't criminalize the harmful and repressive mindset. That change has to come from within each person themselves - until then, we just have to do what we can for the victims, and make it easier for people to escape from such oppressive families.