Well, it could lead to some interesting interactions. Maybe the pureblood families would start to court her in a way, trying to bring her into their circles. But Hermione would probably be very cautious. She might also face some jealousy from those who were once her equals as a Muggle - born but now see her as having an 'advantage'.
She might struggle with her new identity at first. After all, she has always been seen as a Muggle - born. This new discovery could make her question her place in the wizarding world. She may start to explore her pureblood heritage, looking into her family's past and any special magical abilities or traditions that come with it.
One thing that could occur is that Hermione would use this new knowledge to her advantage in the fight against Voldemort. If she has pureblood connections, she could infiltrate certain pureblood - only groups that are sympathetic to Voldemort's cause. She could then gather intelligence and work from the inside to undermine the Dark Lord's plans. She would have to be very clever about it, though, as she doesn't want to lose her integrity or the trust of her friends.