SELECT * FROM tClient WHERE Name1 Like @Name + '%'

<asp:LinqDataSource ID="LinqDataSource1" runat="server" 
        ContextTypeName="DataClassesDataContext" 
        Select="new (Name1, Name2, Name3, BirthDay, DocSerial, DocNumber)" 
        TableName="tClients" OrderBy="Name1, Name2, Name3" 
        AutoGenerateWhereClause="True" 
        <WhereParameters>
                <asp:ControlParameter ControlID="TextBox1" Name="Name1" PropertyName="Text" Type="String" />
       </WhereParameters>
</asp:LinqDataSource>

        LinqDataSource1.Where = "Name1.StartsWith(" + "\"" + TextBox1.Text + "\")";

...
  <script type="text/ecmascript">
    function LookUp()
    {
        CallServer(document.getElementById("TextBox1").value, "");
    }
    function ReceiveServerData(rValue)
    {   
        document.getElementById("WarnSpan").innerHTML = rValue;

    }
  </script>
...
                <asp:TextBox ID="TextBox1" runat="server" Width="210px" onkeyup="LookUp();" ></asp:TextBox>
                <span id="WarnSpan"></span> 
...

public partial class _Default : System.Web.UI.Page, System.Web.UI.ICallbackEventHandler 
{
    protected System.Collections.Specialized.ListDictionary catalog;
    protected String returnValue;

    protected void Page_Load(object sender, EventArgs e)
    {
        String cbReference =  Page.ClientScript.GetCallbackEventReference(this, "arg", "ReceiveServerData", "context");
        String callbackScript;

        callbackScript = "function CallServer(arg, context)" + "{ " + cbReference + ";}";
        Page.ClientScript.RegisterClientScriptBlock(this.GetType(),  "CallServer", callbackScript, true);
    }
    public void RaiseCallbackEvent(String eventArgument)
    {
        if (eventArgument.Length > 4)
        {
            LinqDataSource1.Where = "Name1.StartsWith(" + "\"" + eventArgument + "\")";
            UpdatePanel1.Update();
            returnValue = "";
        }
        else
        {
            returnValue = "Длина строки поиска должна быть больше 4 символов.";
        }
    }
    public String GetCallbackResult()
    {
        return returnValue;
    }
    protected void TextBox1_TextChanged(object sender, EventArgs e)
    {
        LinqDataSource1.Where = "Name1.StartsWith(" + "\"" + TextBox1.Text + "\")";
    }
}

document.getElementById("WarnSpan").innerHTML = rValue;