With CentOS 5.8, which mirrored RHEL 5, the php version available from CentOS repos were less than what was required. Third party repo need to aquire 5.3.x of php with all extensions.
One could acquire from epel repos.
Tip: when setting up that repo make sure it is defaulted to 'disabled' ... before using. It might confuse the system if leaving active. This does mean one needs to enable it on the fly using:
yum --enablerepo=epel list php53* php53-*
One just needs to acquire php53 and extensions ... not all the other stuff that might be available.
BTW, CentOS 5 is now at 5.9.
Consider updating the entire system first before going after php53 with the epel repo.
If ya need it: http://www.thegeekstuff.com/2012/06/enable-epel-repository/
Here's one difference in platform ... with CentOS/RHEL one just needs to acquire php53-mysql (mysqli comes included). As you have discovered, there is no php53-mysqli - separate.
After acquiring, might consider immediately increasing some values in php.ini then. And, apache would have to be restarted to pick up on changes to PHP. ;)
'spirit of sharing', Ken