This answer to another question describes the different values chosen as private key and public key by various Ed25516 implementations. What are the advantages / disadvantages of each?
(Table cropped from diagram in the linked answer, originally taken from "How do Ed5519 keys work?" by Brian Warner.)
To perform an Ed25519 signature operation, you need to know three values, denoted by $\sf RH$, $a$ and $A$ in the diagram. Now, as it happens, these values are not independent:
Thus, all you really need to store is the seed $k$; everything else can be derived from it. Alternatively, it's possible to store $\sf RH$ and $a$ like NaCl does, which saves you the (minor) effort of one SHA-512 hash computation whenever you need to sign something.
There's no particular need to store the public key $A$ as part of the private key, since it can be derived from $a$ and you need to know $a$ anyway to be able to sign. However, deriving $A$ from $a$ requires an elliptic curve multiplication, which is a reasonably expensive operation compared to the other key generation steps. Thus, also storing $A$ as part of the private key provides a modest performance gain compared to storing just $k$ or just $\sf RH$ and $a$.
