A sponge filter is great for a Betta tank.

The Tetra Whisper 10 is a reliable cheap quiet pump. There are others.

Hydro makes a deluxe filter that can use an airstone for more efficiency and less turbulence. Get the smallest one. Many keepers like this filter. It's easy to hide because the sponge lies horizontally. Amazon.com : Elite Single Sponge Filter : Aquarium Filters : Pet Supplies

Use the flexible silicon tubing (thank me later) and a backflow check valve if the pimp is lower than the waterline.

This might answer some questions for you.
www.tropicalfishkeeping.com/betta-fish-bowls-habitats-accessories/sponge-filter-tutorial-318602/http://www.amazon.com/Elite-A900-Single-Sponge-Filter/dp/B0002AQGZQ/ref=sr_1_5?s=pet-supplies&ie=UTF8&qid=1387728412&sr=1-5

 

I run sponge filters in all my tanks. The tetra whisper is awesome. it's not super quiet (no air pump can reasonably be silent), but with the tv on even low, I don't hear it. I have the two sponge filter in my 10, which is probably overkill, but more places for bacteria to colonize is never a bad thing.

 

+1 to everything Hallyx says.
It's the exact set up I use in my 10 gallon tanks the only exception is the I use a double. Just to give you an idea, here are a few photos of my sponge filters in action.

Hopefully these photos will give you an idea of set-up & water movement.

 

Thank you so much, everyone! I'll be buying some supplies soon, with luck!

 

If quiet is an absolute premium, there are some silent power-heads you can hook up to a sponge filter instead of an airstone, but that gets you back into spending money, and it makes more current.

But something to keep in mind if even the faint hum of an air pump is too much.

 

Mike what do you have holding up the tercotta pot????? do you have a HOB on that tank as well as sponge? I use the sponge in my hospital tank jugs but hob on the 6 gallons

 

Just a suction cup in the back where the hole is.
I'm working on DYI as a replacement for the zoo Med Betta log using a terra cotta pot and a suction cup to keep it in place. I'll put up a thread about it as soon as I see how well it works with full instructions. It's soaking in water right now to get the gas bubbles out.

No HOB, just a sponge. It keeps my water sparkling, they are so underrated.

 

thanks def check em out sometime