I was running a 2 x 9 setup on my daughter's bike with 36 and 24 front chainrings and a 7 speed 14-28 tooth rear cassette.
Getting a new rear wheel meant that I could run a 9 speed rear cassette. With an 11-34 tooth gear range it soon because apparent that by going to a single 28 tooth front chainring the gear range would be the same as the 36/24 14-28 range of the previous chainring/cassette combination.
The key advantage is you can do away with having a front shifter, front derailleur and associated cabling. You also do away with a front chainring as well.
You do however need a few extras - without a front derailleur the chain can easily jump off the front chain ring. A chain keeper / guard is needed. Here's a picture of a prototype that I made. It works well. You'll have to excuse the dirty great bolt holding the clamp on, its was just temporary to make sure the concept worked. The picture also shows a 24t granny ring, but the chain keeper profile it optimised for a 28t chain ring.
Here's the final chain keeper (bashring also shown):
A bash ring is also a good idea. It will stop the chain derailling to the outside and also it provides a little added safety by protecting young calves from chain ring 'bite'. I couldn't find a suitable bash ring (30t size just big enough to cover the 110mm spider on the Sugino cranks ) so I custom made one for a perfect fit .
Along with the reduced complexity comes a reduction in weight by going to 1x9. I was able to ditch an XT front shifter pod (120g), a LX front derailleur (145g), 36t middle ring (55g) and some shifter cable and housing (maybe 20g). Total weight of stuff removed - 340g.
But, there was some stuff we needed to add back on: 30t bash ring (30g) and chain keeper (32g).
So total weight saving was around 278g.