The LM317 regulator has a dropout voltage of about 1.5V under light load (suggesting about 10.5V output with 12V input), and your numbers are considerably less, so there seems to be an issue if your load is indeed light.
Possible causes: You should have the capacitors shown in place to prevent possible oscillation. If your input has some ripple that will affect the readings since the regulator would drop out on the troughs of the input waveform. Or possibly measurement error. Since the minimum output voltage has to be approximately 1.25V (not 0.5V) that might be a possibility. Or maybe the schematic you are showing us is not complete (maybe there's a series diode or something like that).
The 240Ω is not a mandatory value, lower is fine, in fact 120Ω is the maximum that meets worst-case Vin-Vout and temperature minimum loading for the LM317. The purpose of having it that particular value is to draw the minimum current required by the regulator when the load is open, while not wasting excessive power.

With the nominal 1.25V Vref it will draw 5.2mA which is typically adequate (especially for Vin-Vout \$\le\$ 20V, but is not really guaranteed. Using 120Ω meets all conditions for the LM317.
The consequence of using too low a value is wasted power. The consequence of using too high a value is that the output voltage may rise above the set value with a light load (usually considered very undesirable behavior). In extreme cases, the voltage may be less stable because the adjustment pin current and current change will affect the voltage more.
If you are making an adjustable supply you would vary the rheostat value along with the resistor in question to get the desired voltage range, based on the formula in the datasheet.