This may sound no-brainer: the session_name() function will have no essential effect if you set session.auto_start to "true" in php.ini . And the obvious explanation is the session already started thus cannot be altered before the session_name() function--wherever it is in the script--is executed, same reason session_name needs to be called before session_start() as documented.

I know it is really not a big deal. But I had a quite hard time before figuring this out, and hope it might be helpful to someone like me.

if you try to name a php session "example.com" it gets converted to "example_com" and everything breaks.

don't use a period in your session name.

Remember, kids--you MUST use session_name() first if you want to use session_set_cookie_params() to, say, change the session timeout. Otherwise it won't work, won't give any error, and nothing in the documentation (that I've seen, anyway) will explain why.

Thanks to brandan of bildungsroman.com who left a note under session_set_cookie_params() explaining this or I'd probably still be throwing my hands up about it.

For those wondering, this function is expensive!

On a script that was executing in a consistent 0.0025 seconds, just the use of session_name("foo") shot my execution time up to ~0.09s. By simply sacrificing session_name("foo"), I sped my script up by roughly 0.09 seconds.

As  Joseph Dalrymple said, adding session_name do slow down a little bit the execution time.
But, what i've observed is that it decreased the fluctuation between requests.
Requests on my script fluctuated between 0,045 and 0,022 seconds. With session_name("myapp"), it goes to 0,050 and 0,045. Not a big deal, but that's a point to note.

For those with problems setting the name, when session.auto_start is set to 1, you need to set the session.name on php.ini!

Hope this is not out of php.net noting scope.

session_name('name') must be set before session_start() because the former changes ini settings and the latter reads them. For the same reason session_set_cookie_params($options) must be set before session_start() as well.

I find it best to do the following.

function is_session_started()
{
    if (php_sapi_name() === 'cli')
        return false;

    if (version_compare(phpversion(), '5.4.0', '>='))
        return session_status() === PHP_SESSION_ACTIVE;

    return session_id() !== '';
}
if (!is_session_started()) {
    session_name($session_name);
    session_set_cookie_params($cookie_options);
    session_start();
}

The description that session_name() gets and/or sets the name of the current session is technically wrong. It does nothing but deal with the value originally supplied by the session.name value within the php.ini file.

Thus:-
$name = session_name();
is functionally equivalent to
$name = ini_get('session.name');
and 
session_name('newname);
is functionally equivalent to
ini_set('session.name','newname');

This also means that:
$old_name = session_name('newname');
is functionally equivalent to
$old_name = ini_set('session.name','newname');

The current value of session.name is not attached to a session until session_start() is called. Once session_start() has used session.name to lookup the session_id() in the cookie data the name becomes irrelevant as all further operations on the session data are keyed by the session_id().

Note that changing session.name while a session is currently active will not update the name in any session cookie. The new name does not take effect until the next call to session_start(), and this requires that the current session, which was created with the previous value for session.name, be closed.

The description has recently been modified to contain the statement "When new session name is supplied, session_name() modifies HTTP cookie". This is not correct as session_name() has never modified any cookie data. A change in session.name does not become effective until session_start() is called, and it is session_start() that creates the cookie if it does not already exist.

See the following bug report for details: https://bugs.php.net/bug.php?id=76413

Always try to set the prefix for your session name attribute to either `__Host-` or `__Secure-` to benefit from Browsers improved security. See https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Set-Cookie#attributes

Also, if you have auto_session enabled, you must set this name in session.name in your config (php.ini, htaccess, etc)