<p>We have a whole bunch of queries that "search" for clients, customers, etc. You can search by first name, email, etc. We're using LIKE statements in the following manner:  </p>

<pre><code>SELECT * 
FROM customer 
WHERE fname LIKE '%someName%'
</code></pre>

<p>Does full-text indexing help in the scenario? We're using SQL Server 2005.</p>

It will depend upon your DBMS.  I believe that most systems will not take advantage of the full-text index unless you use the full-text functions.  (e.g.
MATCH/AGAINST
in mySQL or FREETEXT/CONTAINS in MS SQL)
Here is two good articles on when, why, and how to use full-text indexing in SQL Server:
How To Use SQL Server Full-Text Searching
Solving Complex SQL Problems with Full-Text Indexing