<script type="text/javascript" src="https://code.jquery.com/jquery-3.2.1.min.js"></script>
<select name="nome_cliente" id="" required>
    <option value="" data-evolucao="">Escolha o paciente</option>
    <option value="1" data-evolucao="Evolucao1">Paciente1</option>
    <option value="2" data-evolucao="Evolucao2">Paciente2</option>
    <option value="3" data-evolucao="Evolucao3">Paciente3</option>
</select>
<br><br>
<textarea name="evolucao" readonly>

</textarea>
<!--Jquery-->
<script type="text/javascript">
    $(document).ready(function () {

        $('select[name="nome_cliente"]').on('change', function () {
            var evolucao = $('select[name="nome_cliente"] > option:selected').data('evolucao');

            $('textarea[name="evolucao"]').text(evolucao);
        });

    })
</script>