Do cells phones put out a signal when fully charged through data out? I am wondering this because there is a competition to see who can build the best and most efficient portable cell phone battery backup
Typically, no. The USB charging standard has the Data Pins tied to ground or each other through resistors to indicate what type/capacity charger the phone is plugged into, to let the phone know how much current it can pull. While connecting the phone to a regular USB Host also does not have that information by default (phones using non-standard drivers other than USB Storage might have it, varies by manufacturer).
While phones have charging circuits that are already built for efficiency by switching into trickle charge mode once done charging, you can either passively or actively sense this. One way would be to have your charger measure the current draw, to figure out when the phone went from high speed charging to trickle charge mode. Another is to use bluetooth (preferably Bluetooth 4.0 Low Energy) with a custom app/service that signals the charger when the phone is fully charged.
Both options would, IMHO, require that the phone be unplugged and replugged in (or a button pressed on the charger, or on the app for bluetooth) to start charging again, so that means it would discharge as if it were unplugged. Hope that helps.
If you mean do they put out a radio signal ie connecting to the mobile network when charging via USB, then the answer depends on whether it is switched on or not. When the phone if switched off and is being charged there should be no communication with the network. If you want to make doubly sure, switch it into airplane mode before switching off and charging.
It sounds like what you're building is a power bank, or external recharger battery pack for a (smart)phone. These usually consist of a USB input jack, battery charger, lithium-ion or lithium-polymer battery, battery-to-5VDC regulator, and USB output jack. The phone then gets powered from 5.0VDC on the USB output jack of the power bank, and since the phone acts the same whether this 5V supply comes from wall adapter, laptop, power bank, etc., there is no way to optimize how anything inside the phone performs. The charge time or power supply efficiency inside the phone is completely unaffected by the power bank design. This is assuming the pack isn't underpowered and can put out as much current as the phone would like to have as input.
Presumably what you're trying to optimize is the efficiency of the power regulators inside the power bank. At the least, this would entail choosing switching regulators over linear or LDO regulators for the battery charger and battery-to-5V regulator. From there you could work on optimizing the regulators for the (known) input & output voltage ranges to try to get the converter efficiency as high as you can. You may also need to optimize any battery protection circuitry in the pack so that it wastes as little energy as possible.
As to other features making it "better," you don't specify what this means, but we could guess some possibilities. First would be what size battery you wish to use in the power bank - you might want to design the electronics to work over a broad range of battery sizes, or be highly optimized for one specific size. For the battery itself you can trade off between capacity, power capability, cycle life, size, weight, cost - what chemistry or flavor of lithium-ion do you choose and which cell manufacturer.
Other features might be adding protection against ESD or a short circuit on the output pins. Multiple output USB jacks with individual protection on each. Multiple power input sources such as adding a wall adapter input. How beautiful is the case. In a real product these would be traded off against the price you believe the consumer would be willing to pay.
Mobile phones use Lithium based batteries. Mostly Li-Ion. They are voltage limiting battery. For more on such batteries pl. see http://www.cellphoneshop.net/usbdacaforsa59.html Since we understand that is the voltage that needs to be sensed for perfect charging do see some voltage sensing circuits.
Mobiles, such as old Nokia 100, would give a "fully charged" indication on screen. There is no "data out" as the circuit for data is not connected to charging.
If you see the basic USB connection links pins are 1:Vcc 2: Data+ 3: Data- 4: Ground. Data is isolated from charging points Vcc and gnd to maintain data integrity.
However, for your purpose you may take up the software aspect which indicated that the display "battery full" and use data ports to display it outside the phone. For this understanding of the cell phone model, the battery specifications and the OS running on it are necessary.
More on phone battery here