The following things went wrong in your implementation:

1. Move initialization outside the for loop.
2. You don't need +1 when doing .slice().
3. Return statement return newArr should be outside the for loop.

const search = (nameKey, myArray) => {
    var newArr = [];
  
  for (var i = 0; i < myArray.length; i++) {
    if (myArray[i].name.slice(0, nameKey.length) === nameKey) {
      newArr.push(myArray[i]);
    }
  }
    return newArr;
}

var array = [
  { name:"string123", value:"this", other: "that" },
  { name:"string2", value:"this", other: "that" },
  { name:"string12", value:"ths", other: "that" },
];

var resultObject = search("string12", array);
console.log(resultObject);

Using Array#filter and String#includes:

const arr = [ { name:"string123", value:"this", other: "that" }, { name:"string2", value:"this", other: "that" }, { name:"string12", value:"ths", other: "that" } ];

const res = arr.filter(({ name }) => name.includes('string12'));

console.log(res);

You can use String#startsWith instead of includes if you only need to check if the name starts with the string.

You could do this with Array.filter and startsWith function (docs). If you want to check string12 anywhere in name use includes instead of startsWith

var array = [
    { name:"string123", value:"this", other: "that" },
    { name:"string2", value:"this", other: "that" },
    { name:"string12", value:"ths", other: "that" },
];

const filtered = array.filter(item => item.name.startsWith("string12"));

console.log(filtered);