Set AutoPostback="True"
. This way the event will be fired server-side, and the button will become visible.
<asp:TextBox ID="TextBox1" runat="server" OnTextChanged="TextBox1_TextChanged" OnClick="TextBox1_Click" AutoPostBack="true"></asp:TextBox>
However, if you only want to toogle visility of a button, you really should considerate javascript. This will save a trip back to the server.
<asp:TextBox onclick="txtBox1_ClientClicked()" ID="TextBox1" runat="server" OnClick="TextBox1_Click"></asp:TextBox>
<asp:Button ID="ButtonSearch" runat="server" OnClick="ButtonSearch_Click" Text="Search" style="display:none;" />
<script type="text/javascript">
function txtBox1_ClientClicked(){
var theButton = document.getElementById('<%=ButtonSearch.ClientID%>');
theButton.style.display = 'block';
}
</script>