Eri has been one of My Hero Academia's most enigmatic characters since her introduction. She raised a lot of questions about the specifications of her Quirk and how she managed to be affiliated with the Hassaikai. A few of these mysteries have been resolved since she was rescued from Overhaul's clutches, but one that persists is why she has a horn.
Heteromorphic Quirks aren't at all rare in My Hero Academia. Several characters, both major and minor, have animalistic features -- such as Gang Orca and Hound Dog. These physical traits clearly come from or contribute to their Quirk ability and, for the most part, are accepted as a part of them. Eri's case is a little more complicated because while the horn is linked to her Quirk ability, it is not her Quirk.
Eri's Rewind is one of the most powerful Quirks in the series so far. It allows her to reverse the state of living organisms, however she pleases. As a child, she rewound her father back into nothingness because she had no control over her ability. After experimentation with her Quirk, Overhaul discovered how to use it to create bullets that rewound the Quirk gene in humans. However, now that Eri has control over it, it's been primarily used as a tool for healing. None of these abilities necessitate the presence of her horn.
Her horn is undoubtedly linked to her Quirk though. When she lost control of her Quirk during her rescue from Overhaul, her power continued to stream from her horn. It shrunk after she'd spent all her energy and began to grow larger again as she recovered from the terrifying ordeal. Additionally, when Monoma copied her Quirk, he grew a horn exactly like hers, implying it was directly linked to Eri's Quirk factor.
Right now, it doesn't seem like her horn serves any purpose apart from being a visual indicator of how primed her Quirk is for use. When Monoma copied Rewind, he called it a blank; a term he uses to describe Quriks like One For All and Fat Gum’s Fat Absorption that translate stored energy into their main ability. It implies that Eri can't use Rewind indefinitely and probably has an upper limit to her power. Her horn grows and shrinks depending on how close she is to the peak of her power.
Before she started to learn to control her Quirk, Eri's horn had grown to the largest size it had been yet, and it was putting her in a lot of discomfort. Though unlikely, it's possible that an underuse of her Quirk might have adverse effects on her as well. Whatever the case, Eri's horn does help her and the audience to keep track of the state of her overwhelmingly powerful Quirk.
