LED Driver Choosing help
How efficient a solution with resistors is depends mainly on the input voltage. So your solution with 5&nbsp;V power supply and red LEDs is around 40&nbsp;% efficient (2&nbsp;V over the LED assumed).

Most of the LED drivers I know of, use a single string of series connected LEDs and regulate the current through the string. With 15 LEDs you are looking at around 30&nbsp;V (look up the numbers in the datasheet for the forward voltage). So that is the required output voltage of the driver.

With a 5&nbsp;V input, you need a step up LED driver to make that work. Depending on the driver, the low current you want might be a bit tricky to achieve (around 10 mA). You probably have to calculate the duty cycle of the driver and work out the correct value for the inductor, which might require some fiddling around with the numbers.

The formulas are given in the datasheet usually.
Some drivers come with a spreadsheet which allows you to easily calculate the values.

A lot of the drivers come with a dimming input which works analog or with a PWM. You can use the analog dimming with your idea of the potentiometer.

For example the PAM2841 might fit your bill. But there are hundreds of drivers out there and some are probably even better suited for what you are doing.

With this you might be looking at around 80&nbsp;% efficiency. Beware though, that switch mode power supplies can cause havoc if you do not follow the layout considerations closely. They can be very good at producing electromagnetic interference.