The battery is not the issue. You need to source a DC/DC converter that takes an auxiliary battery pack and provides 5V for a mini-USB plug for smart phones and tablet devices. Of course, laptop computers use a variety of input voltages so a solution for those is not as simple.
There are several DIY battery pack kits out now, and more are on the way. Google Vruzend and NESE as an example.
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These can all be configured as 1S packs of various sizes. That means that they will be 4.2V when fully charged, and 3.3V when low. A DC/DC converter for them will be a "step up" transformer to a voltage higher than 5V (for example, 2S / 8.4V might be useful for this), and as the voltage in the auxiliary pack slowly depletes down to 6.6V, there will be circuitry that adjusts the output to maintain a steady 5V output to the mini-USB plug.
There is no single way to accomplish this, for instance...you can configure the lithium auxiliary pack to be near 12V, because the 12V-to-5V mini-USB chargers are already common. For that method, you might consider using 4S LiFePO4 cells. The problem there is sourcing an affordable charger for the 4S LiFePO4 pack.