Hi there and welcome to Electronics Point
There are two ways to drive the LEDs: directly, or using a boost converter.
A boost converter requires an inductor and generates a higher voltage than the battery voltage, so multiple LEDs can be connected in a string that adds up to more than the battery voltage. This can reduce the number of components and increase the efficiency of the circuit. A self-oscillating boost converter can be made quite simply. It can be turned off fairly easily by a phototransistor when there's light. It's good for between 2 and around 10 LEDs.
Directly powering the LEDs is better if you have a relatively high battery voltage. LEDs can be connected in series strings, with a current limiting resistor for each string. A transistor can be used to enable current to flow through the strings. Significant power can be wasted in the current limiting components.
Adam's design powers the LEDs directly, but uses transistors to limit the current in each string. This wastes significant power and I wouldn't recommend it for a battery-operated circuit.
I did a search for some existing designs:
https://www.google.com/search?q=battery+powered+LED+night+light+circuit&tbm=isch
Then I picked out a few designs that looked good.
Some of them use a nice little trick - a solar cell charges a small rechargeable battery during the day, and also acts as the darkness detector. I've flagged these designs as "rechargeable" in my summaries below.
Here are some interesting results I found:
http://www.electronicshub.org/auto-night-lamp-using-high-power-led/
A simple design that uses an LDR to detect the light level and two transistors to switch the LEDs. Designed to operate from a 12V source.
http://www.instructables.com/id/Electronic-Night-Light/step3/null/
Very simple circuit using an oscillating boost converter, based on a "joule thief" design. The converter is turned OFF in daylight by a phototransistor. The boost converter can generate several times the battery voltage; this would be good if you want a single 1.5V battery to power several LEDs.
http://www.electronicshub.org/battery-powered-portable-light/
A moderately complicated design without nighttime detection (this could be added) using a boost converter and a string of 7 high-brightness LEDs. Intended to run from two CR2032 button cells. The schematic is poorly drawn and the design could be improved.
http://solar.smps.us/solar-lights-outdoor.html
This page includes a little schematic at the bottom that implements a tidy little rechargeable night light using a solar cell and a rechargeable battery. The LED is driven by a boost converter controlled by a Zetex ZXSC310, a 5-pin surface-mounted device that is perfect for this application. It operates from a supply voltage of 0.8~8.0V and delivers constant power to the LED string. Very tidy!
http://www.repairfaq.org/sam/samschem.htm
Search for Brinkmann LED Light. There's a link to the schematic diagram. This is a nice simple self-oscillating boost converter. It doesn't have recharging or night detection but these could be added.
http://members.shaw.ca/NOVOTILL/SolarGardenLight/index.htm
Reverse-engineered schematics for four different commercially manufactured LED night lights (actually they're probably used as outdoor pathway lighting). All designs are rechargeable, and designs 1 and 2 use the solar cell to detect darkness. I recommend circuit 2.
http://www.instructables.com/id/Adding-Oomph-to-the-Garden-Solar-Light/
Another two-transistor oscillating boost converter with rechargeable battery charged from a solar cell and the solar cell used as a dark detector. Some component changes would improve this circuit.
So there are quite a few approaches out there. None of those designs have a 20-minute turn-off feature, but it can be added. If the battery voltage is always 3V or higher, a CD4060 timer IC can be used to measure a 20 minute delay and turn off the LEDs when it expires. Accuracy will not be very good though.
I would definitely recommend the ZXSC310 to drive the LED. It even has a Shutdown input. It's available for less than USD 1.00 from Digikey and for 66p from Farnell in the UK. Alternatively, the two-transistor oscillating boost converter works well enough.
Perhaps you should have a think about what you want and give us some more details. Experiment with some LEDs to find out how many you need, what colour(s) you want, and how much current you want to run them at.
You should add your location to your profile. It can help when we recommend components.
