React, how to access the DOM element in my render function from the same component

You can use ReactDOM.findDOMNode(this) to access the underlying DOM node. But accessing the DOM node and manipulating like you do is against the React style of programming. It's better to use a state variable and called the setState method to re-render the DOM.

Here, no need to use setTimeout. There are lifecycle methods for component, of which componentDidMount is called after the render. So, you can get the reference to your div in the method.

var TodoItem = React.createClass({
  ...
  componentDidMount: function () {
     if(this.myDiv) {
        this.myDiv.style.backgroundColor = "red";
     }
  }
  render:function(){
    return(
        <div className='name' ref = {c => this.myDiv = c}></div>
    );
});

You can make use of ref callback to access the dom element in react, which is what React Docs recommend to follow

and do that in the componentDidMount lifecycle function as refs won't be accessible before the DOM is created

var TodoItem = React.createClass({
    ...
    componentDidMount() {
          setTimeout(function(){
               this.myDiv.style.backgroundColor = "red";
          )}, 1000);
    }
    render:function(){

        return(
            <div className='name' ref={(ele) => this.myDiv = ele}></div>
        )

})

DOCS