The main source for this answer is the Historical Dictionary of Switzerland.
The original name of Switzerland is probably derived from the name of one of the cantons taking part in the legendary "Rütlischwur"-oath, which today is pronounced as "Schwyz" (with ee).
This canton also took part in battles that helped to shape the unified Swiss national identity. For example, they took part in the famous Battle of Sempach in 1386 against the Habsburgs.
Even earlier, there was the Battle of Morgarten, where the soldiers from the canton of "Schwyz" made a lasting impression on southern Germans, who now began to use the terms "Switzer", "Switenses" or "Swicenses" for the whole Swiss cantonal alliance.
During the German Diphthongisation, the pronounciation was altered and, interestingly, the Swiss themselves later took over the new notation but kept the native pronounciation. The source does not mention the transition between "y/i" and [Edit:]"ei" but probably this was also a result of the German Diphthongisation in the middle ages.
To conclude: "Schw-ee-z" is the original pronounciation, which is still used today in Swiss German. But "Schw-eye-z" is used in written language (and in High German) because it was borrowed from the German variant of the term, which underwent the German Diphthongisation.