How can I enable MySQL's slow query log without restarting MySQL?

I followed the instructions here: http://crazytoon.com/2007/07/23/mysql-changing-runtime-variables-with-out-restarting-mysql-server/ but that seems to only set the threshold.

Do I need to do anything else like set the filepath?

According to MySQL's docs

If no file_name value is given for --log-slow-queries, the default name is 
host_name-slow.log. The server creates the file in the data directory unless 
an absolute path name is given to specify a different directory. 

Running

SHOW VARIABLES

doesn't indicate any log file path and I don't see any slow query log file on my server...

EDIT

Looks like I'm using server version 5.0.77, so I needed to do:

SET GLOBAL log_slow_queries = 1;

but I get: ERROR 1238 (HY000): Variable 'log_slow_queries' is a read only variable

I assume I'm going to need to restart the server and have log_slow_queries set in my config?

Try SET GLOBAL slow_query_log = 'ON'; and perhaps FLUSH LOGS;

This assumes you are using MySQL 5.1 or later. If you are using an earlier version, you'll need to restart the server. This is documented in the MySQL Manual. You can configure the log either in the config file or on the command line.

For slow queries on version < 5.1, the following configuration worked for me:

log_slow_queries=/var/log/mysql/slow-query.log
long_query_time=20
log_queries_not_using_indexes=YES

Also note to place it under [mysqld] part of the config file and restart mysqld.

Find log enabled or not?

SHOW VARIABLES LIKE '%log%';

Set the logs:-

SET GLOBAL general_log = 'ON'; 

SET GLOBAL slow_query_log = 'ON'; 

MySQL Manual - slow-query-log-file

This claims that you can run the following to set the slow-log file (5.1.6 onwards):

set global slow_query_log_file = 'path';

The variable slow_query_log just controls whether it is enabled or not.

I think the problem is making sure that MySQL server has the rights to the file and can edit it.

If you can get it to have access to the file, then you can try setting:
SET GLOBAL slow_query_log = 1;

If not, you can always 'reload' the server after changing the configuration file. On linux its usually /etc/init.d/mysql reload