In the past we have always brought our own City Mini (the same most rentals companies use).
As others have stated, most airlines will gate check for free.
One word of advice. We had our City Mini stolen from Downtown Disney outside of Rainforest Cafe. Restaurant management and Disney security were very helpful in trying to track it down, considering I'm sure they deal with this daily. It was never recovered and the only upside was that it was our last night on vacation. Lugging my toddler everywhere was not ideal, especially in the airports.
Our next trip we took a cheaper replacement and hated every minute of it. I cannot say enough about the City Mini, especially it's one-handed folding capability. This is a lifesaver for getting on trams, trains, etc.
So on that note, any recommendations on how to secure your stroller? We looked at renting for our upcoming May trip, however, my wife found a great deal on another City Mini - essentially the same price as renting it for the duration of our next trip. So, we went ahead and purchased another one.
Thoughts?