Oh, thats really, really, really easy. Just identify which of the three bands he was from and call their enrollment office. Sometimes the children/grandchildren of men in our tribe (scandalous men who had kids outside of marriage and what-not) find their way into our enrollment offices where all they have to do is prove kinship to the enrolled tribal member and voila! They fill out their paperwork and complete the other necessary steps for enrollment.
If your grandfather was 1/1, 1/2 or 1/4th Cherokee, it is nearly impossible that he was not enrolled, so no worries there. They'll have his name for sure. Even if you find his name though, the Cherokee have closed their enrollment (unfortunately) because too many people not raised in the culture were taking advantage of the tribe.
The ONLY exception to this...is if he wasn't really Cherokee. Unfortunately, many many families in America have stories of Native American ancestors that aren't exactly true...or they are very exaggerated. Like, some families think that because they see that their great-great grandma lived on Cherokee territory, she must have been Cherokee, when in fact, she wasn't. OR, they Do have a native american ancestor that goes back several generations, but it is so far back that their name, tribe and all the other vital information is lost, so they make something up and their kids believe it and repeat it.
Its kinda like how most american families think that they have Irish, Dutch, English or French ancestry, (which most of them do, even if they don't know for sure) but it still doesn't make them part of Irish, Dutch, English or French cultures because the lineage is so far removed. The difference though, is that most people who think they are Cherokee or some other indian, actually arent. So they create a mental/emotional identity around an idea of something they aren't really a part of, so then they get mad when others point out to them that the possibilities of them actually having that background are very, very slim.