I get a json response from the server that looks something like this:
{
    \"Response\": {
        \"FirstName\": \"John\",
        \"LastName\": \"Smith\",
            
        @adamjyee Your solution works except for nested array of integers. A small fix could be:
function convertKeysToCamelCase (o) {
    if (o === null) {
      return null
    } else if (o === undefined) {
      return undefined
    } else if (typeof o === 'number') {
      return o
    } else if (Array.isArray(o)) {
      return o.map(convertKeysToCamelCase)
    }
    return Object.keys(o).reduce((prev, current) => {
      const newKey = `${current[0].toLowerCase()}${current.slice(1)}`
      if (typeof o[current] === 'object') {
        prev[newKey] = convertKeysToCamelCase(o[current])
      } else {
        prev[newKey] = o[current]
      }
      return prev
    }, {})
[Right to comment but lacking comment priviledge :(]
You would give JSON.parse a reviver function that assigns values to new properties that are lower-cased.
function toCamelCase(key, value) {
  if (value && typeof value === 'object'){
    for (var k in value) {
      if (/^[A-Z]/.test(k) && Object.hasOwnProperty.call(value, k)) {
        value[k.charAt(0).toLowerCase() + k.substring(1)] = value[k];
        delete value[k];
      }
    }
  }
  return value;
}
var parsed = JSON.parse(myjson, toCamelCase);
More information about how it works in this SO answer.
The approach that user '@I Hate Lazy' suggested - using a 'reviver' function is - the right one. However his function didn't work for me.
Perhaps it is because I'm parsing a JSON array. Also I use Resharper and it complained about a code smell :) ('not all code paths return a value'). So I ended up using a function from another SO issue which did work for me:
function camelCaseReviver(key, value) {
    if (value && typeof value === 'object') {
        for (var k in value) {
            if (/^[A-Z]/.test(k) && Object.hasOwnProperty.call(value, k)) {
                value[k.charAt(0).toLowerCase() + k.substring(1)] = value[k];
                delete value[k];
            }
        }
    }
    return value;
}
                                                                        Here is a functional recursive (ES6) approach.
function convertKeysToCamelCase(o) {
  if (o === null || o === undefined) {
    return o;
  } else if (Array.isArray(o)) {
    return o.map(convertKeysToCamelCase);
  }
  return typeof o !== 'object' ? o : Object.keys(o).reduce((prev, current) => {
    const newKey = `${current[0].toLowerCase()}${current.slice(1)}`;
    if (typeof o[current] === 'object') {
      prev[newKey] = convertKeysToCamelCase(o[current]);
    } else {
      prev[newKey] = o[current];
    }
    return prev;
  }, {});
}
// successfully tested input
const o = {
  SomeNum: 1,
  SomeStr: 'a',
  SomeNull: null,
  SomeUndefined: undefined,
  SomeBoolean: true,
  SomeNaN: NaN,
  NestedObject: {
    SomeSentence: 'A is for apple',
    AnotherNested: {
      B: 'is for blahblah'
    }
  },
  NumArray: [1, 2, 3, 4],
  StringArray: ['a', 'b', 'c'],
  BooleanArray: [true, false],
  ArrayOfArrays: [[1,2,], ['a','b']],
  ObjectArray: [{Foo:'bar'}, {Hello:'world', Nested:{In:'deep'}}],
  MixedArray: [1,'a', true, null, undefined, NaN, [{Foo:'bar'}, 'wat']]
}
const output = convertKeysToCamelCase(o);
console.log(output.mixedArray[6][0].foo); // 'bar'
                                                                        You need to write a recursive function that traverses the tree and returns a new tree where the keys in the objects have been updated. The recursive function would call itself to deal with any sub-objects it encounters.