First of all, it's considered bad practice to extend Object.prototype. Instead, provide your feature as stand-alone function, or if you really want to extend a global, provide it as utility function on Object, just like there already are Object.keys, Object.assign, Object.is, ...etc.

I provide here several solutions:

1. Using reduce and Object.keys
2. As (1), in combination with Object.assign
3. Using map and spread syntax instead of reduce
4. Using Object.entries and Object.fromEntries

1. Using reduce and Object.keys

With reduce and Object.keys to implement the desired filter (using ES6 arrow syntax):

Object.filter = (obj, predicate) => 
    Object.keys(obj)
          .filter( key => predicate(obj[key]) )
          .reduce( (res, key) => (res[key] = obj[key], res), {} );

// Example use:
var scores = {
    John: 2, Sarah: 3, Janet: 1
};
var filtered = Object.filter(scores, score => score > 1); 
console.log(filtered);

Note that in the above code predicate must be an inclusion condition (contrary to the exclusion condition the OP used), so that it is in line with how Array.prototype.filter works.

2. As (1), in combination with Object.assign

In the above solution the comma operator is used in the reduce part to return the mutated res object. This could of course be written as two statements instead of one expression, but the latter is more concise. To do it without the comma operator, you could use Object.assign instead, which does return the mutated object:

Object.filter = (obj, predicate) => 
    Object.keys(obj)
          .filter( key => predicate(obj[key]) )
          .reduce( (res, key) => Object.assign(res, { [key]: obj[key] }), {} );

// Example use:
var scores = {
    John: 2, Sarah: 3, Janet: 1
};
var filtered = Object.filter(scores, score => score > 1); 
console.log(filtered);

3. Using map and spread syntax instead of reduce

Here we move the Object.assign call out of the loop, so it is only made once, and pass it the individual keys as separate arguments (using the spread syntax):

Object.filter = (obj, predicate) => 
    Object.assign(...Object.keys(obj)
                    .filter( key => predicate(obj[key]) )
                    .map( key => ({ [key]: obj[key] }) ) );

// Example use:
var scores = {
    John: 2, Sarah: 3, Janet: 1
};
var filtered = Object.filter(scores, score => score > 1); 
console.log(filtered);

4. Using Object.entries and Object.fromEntries

As the solution translates the object to an intermediate array and then converts that back to a plain object, it would be useful to make use of Object.entries (ES2017) and the opposite (i.e. create an object from an array of key/value pairs) with Object.fromEntries (ES2019).

It leads to this "one-liner" method on Object:

Object.filter = (obj, predicate) => 
                  Object.fromEntries(Object.entries(obj).filter(predicate));

// Example use:
var scores = {
    John: 2, Sarah: 3, Janet: 1
};

var filtered = Object.filter(scores, ([name, score]) => score > 1); 
console.log(filtered);

The predicate function gets a key/value pair as argument here, which is a bit different, but allows for more possibilities in the predicate function's logic.