I'm going to advise bringing the stroller with you and not renting one from a rental company. As someone who has traveled numerous times with a child since the age of 1, having that stroller in the airport was not only great but served as comfort item because my little one was used to it. On DD's first trip to Disney, our flight on the way home was delayed for 4 hours.

Without her stroller I probably would have looked a little something like this lovely lady,

. After 2 hours of running around like a loon, she slept for the next 2 in her stroller.
Once you get DS's stroller through security ask for a gate check ticket at your gate. Once you begin to board fold the stroller at the end of the jetway and leave it there; those working for the airline will load it into the belly of the plane. After de-planing, wait in the same location where you dropped it off for it to be brought back to the jetway. No cost is incurred for this. I'd also recommend a stroller bag. It'll prevent the stroller from getting scuffed, etc!