No technology is required on the server. It is quite possible to do a simple search with Javascript.

Search only with jQuery

I made an example using jQuery. The first thing is to have a table with the names and the phones. In my example I did so:

<table id="lista">
    <thead>
        <tr>
            <th><div>Nome</div><div><input id="filtro-nome"/></div></th>
            <th>Telefone</th>
            <th>Ramal</th>
        </tr>
    </thead>
    <tbody>
        <tr>
            <td>Ana</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
        <tr>
            <td>João</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
        <tr>
            <td>Luiz</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
        <tr>
            <td>Mário</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
        <tr>
            <td>Rodrigo</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
        <tr>
            <td>Silvana</td>
            <td>3333-3333</td>
            <td>123</td>
        </tr>
    </tbody>
</table>

Note that I have placed a search field in the header.

With this we can make a script that executes when the filter field is changed. For each line it checks whether the filter matches the name and hides or displays the line as the case may be. Consider the code below:

$('#filtro-nome').keyup(function() {
    var nomeFiltro = $(this).val().toLowerCase();
    $('table tbody').find('tr').each(function() {
        var conteudoCelula = $(this).find('td:first').text();
        var corresponde = conteudoCelula.toLowerCase().indexOf(nomeFiltro) >= 0;
        $(this).css('display', corresponde ? '' : 'none');
    });
});

Demo on jsfiddle

Search only with "pure" Javascript

Since you can not install a server, it may be easier to distribute HTML without any dependency. Thinking about it, I made a version that does not depend on a library like jQuery;

var filtro = document.getElementById('filtro-nome');
var tabela = document.getElementById('lista');
filtro.onkeyup = function() {
    var nomeFiltro = filtro.value;
    for (var i = 1; i < tabela.rows.length; i++) {
        var conteudoCelula = tabela.rows[i].cells[0].innerText;
        var corresponde = conteudoCelula.toLowerCase().indexOf(nomeFiltro) >= 0;
        tabela.rows[i].style.display = corresponde ? '' : 'none';
    }
};

Demo on jsfiddle

Thank you all for the information! With the content passed, I was able to create the following structure:

<table id="lista">
<thead>
    <tr>
        <th><div>Nome</div><div><input id="filtro-nome"/></div></th>
        <th>Telefone</th>
        <th>Ramal</th>
        <th>Celular</th>
        <th><div>Email</div><div><input id="filtro-email"/></th>
    </tr>
</thead>
<tbody>
    <tr>
        <td>Não saber número</td>
        <td>9</td>
        <td>9</td>
        <td>-</td>
        <td>-</td>
    </tr>
    <tr>
        <td>Ocupado</td>
        <td>6</td>
        <td>6</td>
        <td>-</td>
        <td>-</td>
    </tr>
    <tr>
        <td>TI - Sala Servidores - CPD</td>
        <td>*** ****-****</td>
        <td>****</td>
        <td>-</td>
        <td>-</td>
    </tr>

...

And the javascript looks like this:

<script type="text/javascript">

window.onload=function(){

//para nomes
var filtro = document.getElementById('filtro-nome');
var tabela = document.getElementById('lista');
filtro.onkeyup = function() {
    var nomeFiltro = filtro.value;
    for (var i = 1; i < tabela.rows.length; i++) {
        var conteudoCelula = tabela.rows[i].cells[0].innerText;
        var corresponde = conteudoCelula.toLowerCase().indexOf(nomeFiltro) >= 0;
        tabela.rows[i].style.display = corresponde ? '' : 'none';
    }
};

//para email
var filtro2 = document.getElementById('filtro-email');
var tabela2 = document.getElementById('lista');
filtro2.onkeyup = function() {
    var nomeFiltro = filtro2.value;
    for (var i = 1; i < tabela2.rows.length; i++) {
        var conteudoCelula = tabela2.rows[i].cells[0].innerText;
        var corresponde = conteudoCelula.toLowerCase().indexOf(nomeFiltro) >= 0;
        tabela2.rows[i].style.display = corresponde ? '' : 'none';
    }
};

}

</script>

With this, I was able to create an extensive phone book with a search. Many thanks!