The 1950-present hourly and daily data from Fairbanks show some support for the hypothesis, although it's not a strong signal; see the chart below. My analysis looked at all days in June through August with a high temperature at least 10°F above normal, i.e. above 72-84°F depending on the date, and I then took the 25% of these days with the lowest "overnight" (6pm-6am) cloud cover. For each of these 124 days, I found the precipitation in the previous 90 days and plotted this against the overnight decrease in temperature anomaly (i.e. the decrease from high temperature anomaly to low temperature anomaly).
The main feature of the results, I think, is that overnight cooling is usually moderate or large - and rarely small - when the prior 90 days have been either moderately or very wet (above 3" of rainfall). The scatter is large when the prior 90 days have been dry - sometimes there is no cooling relative to normal, but sometimes there is a lot. There is an interesting cluster of large-cooling events at intermediate precipitation levels, but I'm not sure what this means.
Bear in mind that late summer is usually much wetter than early summer in Fairbanks, so most of the points on the left-hand side are from June while most of the points on the right are from July and August. The overall trend is not just reflecting the date, however, because (a) I've used temperature anomalies rather than temperature, and (b) the normal diurnal temperature range actually gets a bit smaller as summer advances in Fairbanks.
One reason for the high degree of scatter in the chart is that overnight wind speed also has an influence on the rate of cooling; calm winds allow the surface layer to decouple so that enhanced low-level cooling can occur. The chart below shows the effect of wind speed for all of the warm summer days regardless of cloud cover.
Finally here's a chart showing the influence of cloud cover regardless of wind speed.