Things that come to mind: Size of the economy of the issuing country, size of population of the issuing country, military power( quantified as defense budget) and country's share in world trade.
Open Markets, Contract Law And An Efficient Legal System
Many places that are unimportant by the population/economy/military size have global currencies. The key is that they tend to have efficient, transparent legal systems and a clear base of contract law. Both the Singapore dollar and the Hong Kong dollar are global currencies, for example, while currencies for the two biggest populations in the world (India and China) are not.
It also helps to have a local advantage in this regard. For example the Dutch guilder was never really a global currency before the Euro came around, because the European slots for that role were dominated by bigger countries like Britain, France and Germany.
To elaborate a bit further, one important prerequisite for some currency $G$ to be "global" is that there be demand for trades between $G$ and a variety of other currencies. That demand really takes off when companies whose native currencies are $A$ and $B$ want to do business. Neither one may be comfortable setting up transactions in the other's country and/or currency, so $G$ provides a mutually acceptable compromise. We then get lots of transactions between $G$ and other currencies.