Does anyone have recommendations for earbuds/earphones that stay in place while running?  While it is a nice distraction from running to keep putting earbuds back in, it does get old after awhile.  Thanks.  

Hmm- guess I am old-school, still using a Sansa mp3 player with tethered earbuds.  

OK. You didn't qualify price point. So here goes.
The BEST solution is from Sensaphonics:   http://www.sensaphonics.com/?cat=10
 
They make silicone custom molded head phones and also make custom molded sleeves for Shure musicians monitors.
Impressions from an audiologist will cost about ($50-75)... well a doctor appt. Get a hearing test while you're there have insurance cover it.
Then the headphones are $450 - 1000.  
 
Now the sleeves are cheaper... $150. Shure SE315 ear buds are $200, so $350.
 
Here's the advantage: they fit PERFECTLY.  You can wear them ALL DAY. You can block out 100% of outside noise at whisper quiet volume. (plug your ears and hum - you get the idea).  They are good for lawn mowing.... in the gym.... on stage (why I have a pair).
 
Downside - on the street - you cannot hear ANYTHING but music and your footsteps.
 
So... this is the BEST solution.