Here is an official Bosnian government source, from the Ministry of Security. In reading it, keep in mind that Cyprus is, as you note in a comment, an EU country. I have added emphasis to the text to call attention to the places where it mentions "EU country":
Visa-free regime for the Multiple Entry Schengen Visas or Residence Permits in the Schengen or EU Member States
Nationals of the countries having a visa-free regime with BiH may enter BiH with their passports without obligation to obtain a visa for entry, exit, transit and stay on the territory of Bosnia and Herzegovina provided they hold a valid multiple entry Schengen visa or a visa of a [sic] EU country or an appropriate residence permit in a Schengen or EU country.
These provisions apply only if a holder of a Schengen visa/EU country visa or of appropriate residence permit in the Schengen area or EU country, arrives to Bosnia and Herzegovina from:
the Schengen area;
a Member State of the European Union;
a country having signed a readmission agreement with BiH, which has entered into force.
In this case, a total stay in BiH cannot exceed 90 days within the 180-day period, starting from the date of first entry, provided that one uninterrupted stay in BiH cannot be longer than 15 days and within a period of validity of the visa or residence permit issued by a Schengen or EU country.