If you go to the subsite, break permissions inheritance and grant the
user Full Control permission level they will have administrative
authority from that site  down in the hierarchy. There will be a few
things they can't do that are reserved for the Site Collection
Administrator or require administrative access to the root level site,
but they will be able to create, modify, and delete permission levels
and groups that they create.

I must be missing something; can't you just give them Full Control on
the Site where you want them to have these permissions?

My previous response seems to have gotten lost. Simply adding the user to
the site  Owners group for the site will give them Full Control over that
site, and that site only. SharePoint already has a group for this purpose.

For the site  that you want the user to have Administrative control over
simply add  them to the Site owners group for that site. This group has full
control over that site.

I did add  the person to the admin group of that site.
But, what happens is that this user cannot change the existing permission groups
for the site.
Thats where i get stumped. I know that i can make the person the admin for the
permission group so that he/she
can have full permissions. But, I was looking for a tweak something like the
mngsiteadmin.aspx file usage
which gives the site  collection admin to people, but specific the site under
consideration.

Site permissions are managed at the root site  of the site collection. If
your user is a member of the Site Owners group for the site you wish them to
manage and they cannot modify those site's permissions, you are still, very
likely, inheriting permissions from the root. Break the permission
inheritance at the site level and you should be good to go.