I have a Baby Jogger Performance series buggy with 20" wheels which I use for running primarily but also for lumpy walks. I have run with DS since he was 4 months old (used a head hugger when he was that tiny) and he loves it - he's 17 months now.
I disagree that it needs to be anything other than what it is intended for (i.e. going into town) - if it is for running then a fixed wheel is essential and by all accounts a permenantly fixed wheel is better than one that you lock (less judder).
It needs to have the biggest possible wheels - 16" as a minimum I would say. Height adjustable handlebar would probably be handy, I find the BJ bar a tad too low (I'm 6ft) and it does take a toll on my shoulders for longer runs.
The back wheels need to be wide enough apart so you can run between them without tripping over them.
Some suspension is good.
Sling style seats are best for absorbing bumps and lumps.
It needs a substantial hood - mainly to protect from sunshine - nothing guaranteed to upset DS more than sun in his eyes!
Stuff that I would like on mine which it doesn't have (and I have added / modded) - reflectors on the wheels, somewhere to mount a light (one forward facing, one backward) and a bell!
I love my Baby Jogger - although it is HUGE it is so easy to push and turn (I wasn't sure how a fixed wheel would be) and it has meant that I have managed much more running than I would if I hadn't had it.
Here endeth my rave