How do I escape a single quote in SQL Server?

How about:

insert into my_table values('hi, my name' + char(39) + 's tim.')

Single quotes are escaped by doubling them up, just as you've shown us in your example. The following SQL illustrates this functionality. I tested it on SQL Server 2008:

DECLARE @my_table TABLE (
    [value] VARCHAR(200)
)

INSERT INTO @my_table VALUES ('hi, my name''s tim.')

SELECT * FROM @my_table

Results

value
==================
hi, my name's tim.

If escaping your single quote with another single quote isn't working for you (like it didn't for one of my recent REPLACE() queries), you can use SET QUOTED_IDENTIFIER OFF before your query, then SET QUOTED_IDENTIFIER ON after your query.

For example

SET QUOTED_IDENTIFIER OFF;

UPDATE TABLE SET NAME = REPLACE(NAME, "'S", "S");

SET QUOTED_IDENTIFIER ON;
-- set OFF then ON again