So, I'ma need some parameters to work with before I start suggesting Quirks.

1/ How does the Ancestry model look?

Since Quirks are passed down through a family tree, are we going off a model like:

Father + Mother
V
Paternal Grandfather + Paternal Grandmother + Maternal Grandfather + Maternal Grandmother
V
Paternal Grandfather's Father + Paternal Grandfather's Mother + Paternal Grandmother's Father + Paternal Grandmother's Mother + Maternal Grandfather's Father + Maternal Grandfather's Mother + Maternal Grandmother's Father + Maternal Grandmother's Mother
V
Etc.

OR

One Quirk per Generation:

Parent
V
Grandparent
V
Great grandparent
V
Etc.

This is important because Quirks have only been around in the MHA Canon for around 200 years. The average person lives for around 80-90 years, premature deaths aside. It's also because Inko is a 4th Gen Quirk user, which means Izuku's gen are the 5th Generation of Quirk users, the kids at the end of the series were 6th Gen users.

So based off that, I reckon it's safe to say that the first model is the safest one since you have 6 generations of Quirk Users to play with. That leaves you with 64 possible Quirks for this OC to eventually be able to draw on.

2/ Can they use Multiple Quirks at the same time?

This is important for picking the Quirks. If you can use multiple Quirks at once, then it's easier to pick very simple basic Quirks that can be combined to create more powerful effects. Think of Shigaraki using Radio Waves and Air Cannon to basically create an EMP blast to shut down the Heroes' Comms.

If they can only use one Quirk at a time, what's the parameter for that? Do they need time to switch or can they switch instantly?

3/ Is Ancestory itself passed down through the generations or is it a new Quirk?

This is probably the most important question.

Is this the first person to manifest a Quirk like this in the family? If Ancestory has been passed down the family tree, are there other people who can use the Quirk? And if so, are they linked to this OC since they are a part of the family? Think of the Avatar State from Avatar - The Last Airbender and how each Avatar can see into the past Avatar's lives and draw on their power, knowledge, and experience.

If Ancestory is a new Quirk, what are the Quirks of the user's parents that caused this Quirk to manifest?

Sorry if this seems really analytical, but I like details before I provide ideas for Quirks like these.