In 1901 which Big Ten member became the first non-Ivy League school to win a National Championship?

Michigan.

Except that they were not a member of the Big Ten at that time: it was the "Western Conference." And there were nine schools in it. Also, they didn't actually win a national championship in 1901; they "won" it 40 years later when the Helms Foundation retroactively selected them as 1901 champion.