Using Services, request new password
Services user update function internally calls Drupal's user_account_form function (via user_profile_form).  In user_account_form, it's specifically looking for pass-reset-token as a $_GET parameter, so to update password via Service, unfortunately you have to pass it as a GET parameter on a PUT call.

Example:

PUT https://drupal/:api-endpoint:/user/:uid:?pass-reset-token=:pass_reset_token: 
and the body is {"pass":":new-password"}.

If you wanted to be able to pass "pass_reset_token" as a parameter in the body, you could override _user_resource_update so that it checks if the pass_reset_token parameter is present and puts it into $_GET['pass-reset-token'] prior to calling user_profile_form.