How can my six year old son, who only takes 2 units of Lente in the morning, wake up 24 hours later with a blood sugar of 61 mg/dl [3.4 mmol/L]?

The reasons may have multiple, possibly even interacting, explanations including reasons pertaining to meal planning, evening activities, duration of diabetes and “the honeymoon”. You should take this concern to your son’s pediatric diabetes team to see if his meal/insulin regimen needs to be adjusted.

There are several possible means of “attacking” this:

Simply add more protein or a long lasting carb (like an Extend Bar) with the bedtime snack. This is easy and inexpensive. However, this presumes that the morning lower readings are relatively frequent. I like adding sugar-free Carnation Instant Breakfast with milk for the bedtime snack: increased protein for little extra carbs.
Consider a switch to NPH, rather than Lente. Lente lasts a bit longer than NPH.
Consider a switch to Ultralente. While Ultralente lasts longer than Lente, it tends not to “peak” as high.

DS