FRANKFURT—Europe’s faltering economy and fractious politics are complicating the region’s biggest personnel question in years: Who will replace Mario Draghi as president of the European Central Bank?

Mr. Draghi, who previously ran Italy’s central bank, emerged as the dominant force in the European economy and financial markets during his seven-plus years as ECB head, holding the currency zone together by breaking taboos, including with the launch of a massive bond-buying program.