I'd go for something in between 1 and 2, for the reasons David mentioned:

You do not want to expose the ID of things unless necessary.

However, exposing the ID might become necessary at some point in time. If that happens, backwards compatibility is a concern. So, i would do this:

{
    "id": 1,
    "firstName": "Jane",
    "lastName": "Doe",
    "active": true,
    "gender": {
        "name": "FEMALE"
    },
    "status": {
        "name": "FULL_TIME"
    }
}

That has the same properties as option 2 has; but it has the benefit that adding the ID later on does not introduce a BC break:

{
    "id": 1,
    "firstName": "Jane",
    "lastName": "Doe",
    "active": true,
    "gender": {
        "id": 1,
        "name": "FEMALE"
    },
    "status": {
        "id": 3,
        "name": "FULL_TIME"
    }
}

I'd go for something in between 1 and 2, for the reasons David mentioned:

You do not want to expose the ID of things unless necessary.

However, exposing the ID might become necessary at some point in time. If that happens, backwards compatibility is a concern. So, i would do this:

{
    "id": 1,
    "firstName": "Jane",
    "lastName": "Doe",
    "active": true,
    "gender": {
        "name": "FEMALE"
    },
    "status": {
        "name": "FULL_TIME"
    }
}

That has the same properties as option 2 has; but it has the benefit that adding the ID later on does not introduce a BC break:

{
    "id": 1,
    "firstName": "Jane",
    "lastName": "Doe",
    "active": true,
    "gender": {
        "id": 1,
        "name": "FEMALE"
    },
    "status": {
        "id": 3,
        "name": "FULL_TIME"
    }
}

As Eric points out in the comments, this still uses the entity name as a uniqe identifier. If the ID is introduced later, the name must still remain the same because older clients could (or rather will) have coded towards it.