The answer up front
Take the short version and save yourself a week. Bosnia and Herzegovina takes a simpler single-step certificate, so a consulate never enters this route. Put another way, consular legalization for Bosnia and Herzegovina is not a stage on your file. One official mark in Italy is the whole of the official work.
That matters most for family records: a birth entry, a marriage entry, a death entry, a certificate of no impediment. Those are the papers people carry along this corridor most often, and they all follow the same short path. Ask the receiving office whether a recent re-issue is wanted, because that is a cheap question with an expensive answer if you skip it.
Which office in Italy marks it
Two offices do the marking, and which one depends on who put their name to the paper. The Prefettura is the state's provincial office, and it handles records issued by public bodies, from a town hall entry to a state school report. The Procura della Repubblica is the prosecuting office kept at the court, and it handles court decisions and papers signed before a notary.
So when people ask us how to legalize documents for Bosnia and Herzegovina, our first question is about the signature, not the country. Send a scan and we can say in a minute which of the two counters is right. A paper sent to the wrong one comes back untouched, and that is a fortnight gone.
Asking the right body
Write to the body that will actually read the paper and ask which mark it wants on it. A university office, a registry, an employer and a court may each answer a little differently. That single reply is what settles document legalization for Bosnia and Herzegovina in your case. Print it, keep it with the file, and produce it if anyone at a counter hesitates.
You may still meet a form that asks for embassy attestation for Bosnia and Herzegovina, especially an old one. Do not follow it blindly, and do not ignore it either: ask the body that issued the form whether the line is still live. If they confirm it, forward their words to us. That is the one case in which embassy legalization for Bosnia and Herzegovina from Verona would come back onto the table, priced as its own job.