Set state in life cycle events from props
In order to keep to the React pattern and stick to immutability of props simply separate your code to have a parent component which carries out the logic and passes the the date to the child component which can then be displayed in the child's render component. 

It would then look something like this:

var React = require('React');

var MomentDisplay = React.createClass({
    propTypes: {
        date: React.PropTypes.string.isRequired
    },
    render: function() {
        return (
            &lt;span&gt;
                {this.props.displayDate}
            &lt;/span&gt;
        )
    }
});

var MomentAgo = React.createClass({
    getInitialState: function () {
        return {
            date: null,
            displayDate: null
        };
    },
    componentWillMount: function () {
        this.setState({date: this.props.date});
    },
    componentDidMount: function () {
        setTimeout(function(){
            this.setState({displayDate: date.format('DD MMMM YYYY')});
        }, 3000);
    },
    render: function () {
        return (
            &lt;MomentDisplay displayDate={this.state.displayDate} /&gt;
            )
        }
});