Storing database connection in a session variable

You can't store database connections or result sets in the session, since those are resources, and:

Some types of data can not be serialized thus stored in sessions. It includes resource variables or objects with circular references (i.e. objects which passes a reference to itself to another object).

http://php.net/manual/en/intro.session.php

You can extract a result set into a normal array and store that in the session like any other variable. That would be a fairly typical use case for sessions anyway. Just be careful not to store too much data in the session, as that can become more taxing than fetching it from the database.

Even if you could do this (resource vs. data), this is a bad idea. You'll wind up with lots of concurrent open connections, which will blow your max connections very quickly... especially if its lifecycle is expanded beyond sub 100ms (depending on your queries) to 20 minutes or more. With open connections, something like MySQL also won't be able to reset its memory allocations properly, and the whole system sort of goes to hell. In short, this is not what DBs are for unless the only consumer of your code will be a single user.

As an alternative, I'd highly recommend caching technologies which are designed specifically to reduce database load and obviate connection times. Using something like, at its simplest, memcached will dramatically improve performance all the way around, and you'll be able to specify exactly how many system resources go into the cache -- while letting the database do its job of getting data when it needs to.