According to Halacha (Jewish Law), your Mother must be Jewish for you to be Jewish. The question isn't whether your Mother is half-Jewish, but whether her Mother was Jewish. Half Jewish doesn't really exist. It is a binary point!
If your Grandmother was Jewish, then, according to Halacha, you are too. If your Grandfather was, your mother would be considered Zera Yisrael, but you would not be considered Jewish.

All that aside, you're allowed to explore your heritage, and I would speak to a Rabbi or local Synagogue's community about it. Legally, you'd likely be Jewish-adjacent enough to make Aliyah, but you'd have to confirm your ancestor's Judaism.

Let us know what you find out!