tl;dr: The number is often given as 252 +/- 3 K (or about -21°C or -6°F), so @BowlOfRed pretty much nailed it using first principles!
Without atmosphere, equilibrium temperature at 1AU is about -17C. I would expect that to be the temp "near" the surface around the equator. Colder as the latitude increases.
Reading the references, it seems that the diurnal oscillation (what Earthies might call "monthly" or Lunies might call "daily") in temperature decays with a 1/e depth of about 30 centimeters, so by the time one reaches 1 meter, the temperature "stabilizes" temporally, but of course will increase with depth since the Moon's interior is hot. You can see this in Figure 2 of the last paper ("Subsurface temperature histories, covering 3.5 years...") where the oscillations in temperature at different depths along the probe are plotted.
Further, some of this data can already be found in @KimHolder's question Does the temperature near the surface of the Moon rise rather quickly with depth? who quotes as NASA News article Down the Lunar Rabbit-hole which then quotes Mark Robinson of Arizona State University, principal investigator for the LRO camera.
"The tunnels offer a perfect radiation shield and a very benign thermal environment," says Robinson. "Once you get down to 2 meters under the surface of the Moon, the temperature remains fairly constant, probably around -30 to -40 degrees C."
Why the difference between the "ground truth" actual temperature measurements discussed below and the quote of a quote?
(image source also here) My guess is that the -30 to -40 number is a mid-latitude number, while the following concentrates on nearer to equator values. There's no physical way the temperature can drop by 10 or 20C in 1 meter of increasing depth, so the difference is more context and location.
I'm not an expert on planetary science, but the oft-cited number for an average sub-surface temperature about 1 meter below the surface seems to come from the following papers:
- The Apollo 15 lunar heat-flow measurement, Langseth, M.G., Clark, S.P., Chute, J.L. et al. The Moon (1972) 4: 390. https://doi.org/10.1007/BF00562006
The heat-flow experiment is one of the Apollo Lunar Surface Experiment Package (ALSEP) instruments that was emplaced on the lunar surface on Apollo 15. This experiment is designed to make temperature and thermal property measurements in the lunar subsurface so as to determine the rate of heat loss from the lunar interior through the surface. About 45 days (1 1/2 lunations) of data has been analyzed in a preliminary way. This analysis indicates that the vertical heat flow through the regolith at one probe site is 3.3 × 10−6 W/cm2 (±15%). This value is approximately one-half the Earth's average heat flow. Further analysis of data over several lunations is required to demonstrate that this value is representative of the heat flow at the Hadley Rille site. The mean subsurface temperature at a depth of 1 m is approximately 252.4K at one probe site and 250.7K at the other. These temperatures are approximately 35K above the mean surface temperature and indicate that conductivity in the surficial layer of the Moon is highly temperature dependent. Between 1 and 1.5m, the rate of temperature increase as a function of depth is 1.75K/m (±2%) at the probe 1 site. In situ measurements indicate that the thermal conductivity of the regolith increases with depth. Thermal-conductivity values between 1.4 × 10−4 and 2.5 × 10−4 W/cm K were determined; these values are a factor of 7 to 10 greater than the values of the surface conductivity. If the observed heat flow at Hadley Base is representative of the moonwide rate of heat loss (an assumption which is not fully justified at this time), it would imply that overall radioactive heat production in the Moon is greater than in classes of meteorites that have formed the basis of Earth and Moon bulk composition models in the past.
- In-situ measurements of lunar heat flow (also here) in Soviet-American Conference on Geochemistry of the Moon and Planets, Langseth, M. G., and S. J. Keihm, NASA Spec. Publ., NASA SP-30, 283 – 293, 1977.
The number is often given as 252 +/- 3 K (or about -21°C or -6°F) as for example in Latitude variation of the subsurface lunar temperature: Lunar Prospector thermal neutrons R. C. Little W. C. Feldman S. Maurice I. Genetay D. J. Lawrence S. L. Lawson O. Gasnault B. L. Barraclough R. C. Elphic T. H. Prettyman A. B. Binder
First published: 29 May 2003 https://doi.org/10.1029/2001JE001497
Some of the experimental and theoretical details behind the data and its analysis can be read in Revised lunar heat-flow values, Langseth, M. G., Keihm, S. J., & Peters, K., Lunar Science Conference, 7th, Houston, Tex., March 15-19, 1976, Proceedings. Volume 3. (A77-34651 15-91) New York, Pergamon Press, Inc., 1976, p. 3143-3171.