[typescript] How do I prevent the error "Index signature of object type implicitly has an 'any' type" when compiling typescript with noImplicitAny flag enabled?

There is no need to use an ObjectIndexer<T>, or change the interface of the original object (like suggested in most of the other answers). You can simply narrow the options for key to the ones that are of type string using the following:

type KeysMatching<T, V> = { [K in keyof T]: T[K] extends V ? K : never }[keyof T];

This great solution comes from an answer to a related question here.

Like that you narrow to keys inside T that hold V values. So in your case to to limit to string you would do:

type KeysMatching<ISomeObject, string>;

In your example:

interface ISomeObject {
    firstKey:   string;
    secondKey:  string;
    thirdKey:   string;
}

let someObject: ISomeObject = {
    firstKey:   'firstValue',
    secondKey:  'secondValue',
    thirdKey:   'thirdValue'
};

let key: KeysMatching<SomeObject, string> = 'secondKey';

// secondValue narrowed to string    
let secondValue = someObject[key];

The advantage is that your ISomeObject could now even hold mixed types, and you can anyway narrow the key to string values only, keys of other value types will be considered invalid. To illustrate:

interface ISomeObject {
    firstKey:   string;
    secondKey:  string;
    thirdKey:   string;
    fourthKey:  boolean;
}

let someObject: ISomeObject = {
    firstKey:   'firstValue',
    secondKey:  'secondValue',
    thirdKey:   'thirdValue'
    fourthKey:   true
};


// Type '"fourthKey"' is not assignable to type 'KeysMatching<ISomeObject, string>'.(2322)
let otherKey: KeysMatching<SomeOtherObject, string> = 'fourthKey';

let fourthValue = someOtherObject[otherKey];

You find this example in this playground.