Tecumseh had a relationship with a white woman named Rebecca. What was her last name and what became of her?

Her surname was Galloway.

Tecumseh became a great orator and maintained a great influence among tribes from Ohio to Florida. He was often in Xenia and maintained a close relationship with the James Galloway family. He fell in love with Rebecca Galloway. However they never married because he would not take up the white man's ways and she would not become an Indian."

https://owlcation.com/humanities/Tecumseh-Indian-Chief-Warrior-and-Nation-Builder
publications.newberry.org/indiansofthemidwest/indian-imagery/how-we-know/video-transcript-tecumseh-imagery no longer exists

Rebecca Galloway - "Tecumseh... in time asked the girl's father if he could marry her. Mr. Galloway respected Tecumseh and left the decision up to his daughter. Rebecca said she would marry Tecumseh if he would give up his Indian traditions and live with her as a white man. Tecumseh pondered the decision for a month, and finally he told Rebecca that he could never abandon his people. With that, he bade good-bye to his white friends, and thereafter Tecumseh never took a wife, either Indian or white."

http://www.trivia-library.com/a/famous-native-americans-tecumseh-part-1.htm

Rebecca Galloway taught Tecumseh to speak English and to read the Bible, Shakespeare, and history books. When he proposed marriage, she refused unless he would take up the life of a white man. That he could not do.

I am unaware of any record of what became of Rebecca. Later, when Tecumseh was challenged by Wm. Henry Harrison to make the sun stand still, he correctly predicted an eclipse, pretending to cause it. Possibly he learned of eclipses from his studies with Rebecca.

http://www.trivia-library.com/a/famous-native-americans-tecumseh-part-1.htm

Bet you watched that lunar eclipse last night, eh?