By coincidence I was looking for a bycycle dynamo operated LED lamp. White LEDs require about 3.6V, but because the supply varies from the speed you cycle (6V max, AC) I had to put a bunch of electronics in there for it anyway. Furthermore I wished high effiency, I don't want to be cycling for a resistor converting my power into heat.. So I started looking.
Also note that if you get a 1.4V battery, if you drain it, the voltage will slowly drop to about 1V. At 0.8V it's completely dead. You may want to consider to have a working product at 1.2V or 1.1V for example, also if you want to support rechargeable batteries.
For that you really need a DC/DC converter to boost a voltage up. The LT1932 is probably even more suitable for your purpose. It converts a low voltage to a constant current (which you need for a LED). It's a bit expensive (because 1) it's linear technologies, 2) you're trying to do something low-voltage), but it is able to drive a single LED from 1V. It also has a SHDN pin so you can control it:

It can drive several white LEDs (they require over 3V drop each) from 2V input. This figure shows 4 white LEDs, so that's why it needs 2.7V minimal. I don't know how it will behave if you put only 1 LED in there, but I think it will work just fine.
All you need for this driver is shown there. Rset sets the current through the LEDs(in the datasheet is probably a table). It drives the LEDs in this example with 15mA. And as said, LEDs are controlled by current not by voltage. The resistor you normally use only sets a 'fixed' current (for a certain voltage you apply on the system). This regulator is set to a certain current with the resistor Rset, and then you're done. If you put another LED in series, it will adjust the voltage so the current stays the same. Ofcourse, this has limits, but you won't reach that I suppose.
There are more of these IC's and are quite handy. You probably find more examples that are cheaper, but might not be able to work from 1V.
You can use a microcontroller to control a digitally-controlled potentiometer... there are lots to choose from. They vary in their control mechanisms, but there are several available that can be controlled over I2C or SPI.
The MCP4151-103E/P looks to be cheap, through-hole, pretty capable, and is controlled over SPI for instance.
Best Answer
Theoretically, yes you could use a pot to control the brightness of an LED. In practice, not so much.
To start with, let's assume that the LED has a \$V_F\$ of 2.0v, an \$I_F\$ of 20 mA, and our power supply has 5v. If we wanted a standard current limiting resistor it would have to be 150 ohms to limit the current to 20 mA.
With a pot, we also want a 150 ohm fixed resistor in series. The reason for this is that the pot will go down to 0 ohms, and we don't want to blow anything up in that case. So by putting the 150 ohm resistor in there there will be a maximum current of 20 mA through the LED.
Let's also say that we want the LED current to go down to 1 mA. Unless the pot has a super high resistance, it won't go down to 0 mA, and 1 mA seems like a reasonable lower limit. To make that work, our pot needs to be about 2K Ohms.
Going through the math, the maximum power dissipation on the pot is when it is at about 8%, and the resistance is 160 ohms. In this case the dissipation in the pot is about 0.016 watts -- which is fine for almost every pot. Even so, it is an important step to make sure you won't be burning up your pot.
But here is the important thing: The human eye has a logarithmic response to brightness. Let's say that we have 100% power going through the LED and we want to turn it down. It needs to go down to about 50% before we sense that as being reasonable. The next step down would be at 25%, etc.
Put a different way, if our knob was marked 1 to 10, then 10 would be 100%, 9 would be 50%, 8=25%, 7=12%, 6=6%, 5=3%, etc.
The problem is that a standard pot doesn't quite do that. It will work, and the LED will be dimmed. But a large part of the pots range (maybe 50%) will essentially be useless, producing very little change in brightness.
You might be able to use an audio pot, which has a logarithmic taper, but I'm guessing that the log part is in the wrong direction. (Sorry, even though I work in audio I don't use log taper pots.)
So yes, you can use a pot. It just might not give you the effect you seek.