Why not:

onItemClick: function (event) {

    event.currentTarget.style.backgroundColor = '#ccc';

},

render: function() {
    return (
        <div>
            <ul>
                <li onClick={this.onItemClick}>Component 1</li>
            </ul>
        </div>
    );
}

And if you want to be more React-ive about it, you might want to set the selected item as state of its containing React component, then reference that state to determine the item's color within render:

onItemClick: function (event) {

    this.setState({ selectedItem: event.currentTarget.dataset.id });
    //where 'id' =  whatever suffix you give the data-* li attribute
},

render: function() {
    return (
        <div>
            <ul>
                <li onClick={this.onItemClick} data-id="1" className={this.state.selectedItem == 1 ? "on" : "off"}>Component 1</li>
                <li onClick={this.onItemClick} data-id="2" className={this.state.selectedItem == 2 ? "on" : "off"}>Component 2</li>
                <li onClick={this.onItemClick} data-id="3" className={this.state.selectedItem == 3 ? "on" : "off"}>Component 3</li>
            </ul>
        </div>
    );
},

You'd want to put those <li>s into a loop, and you need to make the li.on and li.off styles set your background-color.

Two ways I can think of are

var TestApp = React.createClass({
    getComponent: function(index) {
        $(this.getDOMNode()).find('li:nth-child(' + index + ')').css({
            'background-color': '#ccc'
        });
    },
    render: function() {
        return (
            <div>
              <ul>
                <li onClick={this.getComponent.bind(this, 1)}>Component 1</li>
                <li onClick={this.getComponent.bind(this, 2)}>Component 2</li>
                <li onClick={this.getComponent.bind(this, 3)}>Component 3</li>
              </ul>
            </div>
        );
    }
});
React.renderComponent(<TestApp /> , document.getElementById('soln1'));

This is my personal favorite.

var ListItem = React.createClass({
    getInitialState: function() {
        return {
            isSelected: false
        };
    },
    handleClick: function() {
        this.setState({
            isSelected: true
        })
    },
    render: function() {
        var isSelected = this.state.isSelected;
        var style = {
            'background-color': ''
        };
        if (isSelected) {
            style = {
                'background-color': '#ccc'
            };
        }
        return (
            <li onClick={this.handleClick} style={style}>{this.props.content}</li>
        );
    }
});

var TestApp2 = React.createClass({
    getComponent: function(index) {
        $(this.getDOMNode()).find('li:nth-child(' + index + ')').css({
            'background-color': '#ccc'
        });
    },
    render: function() {
        return (
            <div>
             <ul>
              <ListItem content="Component 1" />
              <ListItem content="Component 2" />
              <ListItem content="Component 3" />
             </ul>
            </div>
        );
    }
});
React.renderComponent(<TestApp2 /> , document.getElementById('soln2'));

Here is a DEMO

I hope this helps.

Here is how you define a react onClick event handler, which was answering the question title... using es6 syntax

import React, { Component } from 'react';

export default class Test extends Component {
  handleClick(e) {
    e.preventDefault()
    console.log(e.target)
  }

  render() {
    return (
      <a href='#' onClick={e => this.handleClick(e)}>click me</a>
    )
  }
}