EdDSA (Ed25519) is any random number sufficient for a good private key?
Yes.

In order to understand why, you need to understand how the public key is computed. 

The secret key is a scalar. A fixed base point is multiplied by that scalar, and no matter what that scalar value is, you will always end up with a point in the same group as the base point.

You may want to avoid an all-zero secret key but the key space is so large that if the scalar comes from a PRG, this is not something you have to worry about.

The order of the group is 2^252 + 27742317777372353535851937790883648493. The secret should be considered modulo that value. Any secret key between 0 and that value will produce a unique public key.

The secret key can be a uniform value in that range but applications usually just generate a 256 bit key, which doesn't make any practical difference.

Note that most Ed25519 implementations mask the lower 3 bits ("clamping"). So 1 will produce the same public key as 2 or 3. This is a simple way to prevent small-subgroup attacks.
But once again, most applications usually just generate 256 random bits, and don't really have to care about this.