Basically, I am fairly new to Arduinos. I created a project that I have started using daily, and some friends have asked me if I could make them one too. I am looking to make the project cheaper so it is easier to produce. Right now, the project uses an Arduino UNO to control an LCD screen and receives input from 5 buttons and a reed switch. It uses all digital ports on the arduino. I have looked at cheaper arduinos like the Teensy, but I just can't find one less than 10 dollars. I see things like the ATTiny, but I can't see how that could work with both an LCD screen and 6 inputs. Any suggestions that the forum has would be helpful.
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If you're ready to get into the codes and adapt it for your application, you have some options indeed.
An ATTiny is a good idea. If you're refreshing your LCD screen slowly, you can use serial to parallel chips like the 74HC595, a simple shift register.
There are also ways of reading several button states with only one analog input (google "analog button input").
If you at an ATTiny doesn't suit your needs, you can always make your own board using a MCU and some components. Arduino board are good for breadboarding and usable out of the box. But for specific application, you can always make your own board with only the features you need: that'll peobably be cheaper.
A last thing: try different suppliers for your parts. You can find good deals online, that's worth at least a try.
...and some friends have asked me if I could make them one too.
I'd ask them to kick-in some money for the parts. 
It's one thing to say, "It's cool and I'd like to have one", and it's another thing to say, "I want one and I want to buy it". They will appreciate it more if they have to pay for it! And, I assume you'd still be donating several hours of labor.
If you want to do it because it's fun, or because you'd rather spend time than money, fine...
But building it "from scratch", you can run into unexpected expenses, and you can end-up burning-up parts or buying the wrong parts, or there's something you didn't think of, and it usually doesn't come-out as cheaply as you expect...
I've built a few simple Arduino projects and by the time I get it into a box with a power supply, connectors, switches, and whatever miscellaneous "cheap" electronic parts I need, I'm usually over $100 USD. (I usually leave the other electronics on a breadboard, so that adds to the cost too.)
One of the coolest things about the Arduino is that it's ready-to-go on a circuit board. I wouldn't build a custom board unless I needed maybe 10 units or more, or if I needed Personally, to make it smaller, or for some other non-economic reason. And in that case, I'd make a custom PC board. Even if I was making a custom PCB for the other circuitry, my first preference would be to use the pre-built Arduino board for the microcontroller.
You could get a batch of your own cards made, $9.90 plus shipping from itead studio for 5cm x 5cm cards, with pads/holes that make connecting to your other components easy, or mount them on the board directly.
Sounds like a pretty straightforward design. Does it use USB port after the sketch is installed? If not, you can just have an ICSP header for bootloading and installing the sketch - or a 2nd header for sketch download after bootloading, and keep the USB/Serial function off board.
I could adapt any of my existing designs for you, probably take a 1/2 hour tops for the few components needed. Are you handy with a soldering iron for assembling cards?
Oh, they definitely will be paying for it, but I want to bring the price down for them, because right now its at about $60 not including labor, so I am hoping to bring it down to the 30 range. The ATTiny seems the best option now that I know you can use 1 analog input for multiple digital buttons states. I will google that and see where I end up. I am definitely thinking of getting some custom-printed boards, but I want to make a final design before I do that. I don't need the USB after I upload the sketch, so an using an ATTiny to make my own board seems to be the cheapest option.
nano, has usb,
costs $3.27
hard to get cheaper than that to play with.
dave-in-nj:
Geekcreit® atmega328p nano v3 controller board improved version module development board Sale - Banggood.comnano, has usb,
costs $3.27hard to get cheaper than that to play with.
Wow, that is cheap, that would be perfect and much easier to implement (with my limited electronics knowledge).
The cheapo pro-mini and nano clones are absolutely the way to go if you're just making a few of them, and they don't need to look particularly pro or anything.
The nano clones usually use the CH340G for USB, not the FTDI chip like the real nano, or the 16u2 like the uno, but these still work fine.
With all the clones, you should expect to have to bootload them (regardless of what they say about having bootloader), and don't assume they're working at all when you get them. When you get a $3 mcu board, the quality is variable.