Ranger,

"LIKE" requires the wildcard character % to function the way you want it to.

For example,
SELECT *
FROM tblPublisher
WHERE City LIKE 'New %'

... would select all rows where the city starts with "New ".

The SQL in your example would omit the %.

I think you want something like the following (although I am using SQL Server 2005, so I can't test the 2000 code).

declare @LastName varchar(63);
declare @sql varchar(255);

set @adName = 'Jo';

-- and your Lastname line could be
SET @sql = @sql + ' AND Lastname LIKE ' + char(39) + @Lastname + '%' + char(39)

SET works better than SELECT when assigning values to variables.