We've designed a php/mysql e-commerce website built on open source
software for a client and they are claiming the site speed is too
slow.  What are best strategies for improving site speed?

try some stress tests
You can use some automated test software such as Visual Test.

One method is Queuing
- having multiple 'strands' attack one database is sure to slow it down
- like multiple similar threads slow an operation

Another is replication, 3 layers, top level diverts requests to
multiple machines, the second layer contains identical copies, the 3rd
layer contains the core data, it gets all update stuff and pushes it
up to the second layer.
(I suspect Gurgle works like that)

That works where requests for data are far greater than data writes.

Single function and single user machines are far more efficient than
those serving multiple clients
- one can restrict the number of clients to two for the processing
(data) intensive beasts - output and high priority input.

At the crudest, make the web pages point to one IP for Images, another
for Static pages and a third where anything needs to be done.

Build a pyramid of machines - physically similar - and you'll see what I mean
- stick the pyramid on a smart router and it will sing.

1) Make sure you have selected a good hosting company (External factor)

2) Make sure there are no HTML errors in the website. This also makes
the site search engine friendly.

3) Remove tables and allow strict XHTML files diven by external CSS
templates. Using XHTML Strict standards should also minimize your
bandwidth usage because the size of the file is shrunk tremendouly
when CSS is used.
   Sources: http://www.nypl.org/styleguide/
            http://www.alistapart.com/articles/seo/

4) If you have too much JavaScript make them driven externally. See
http://www.hypergurl.com/jsfiles.html for instructions.

I hope this helps.