If Leona, Zealot lowers a unit's might from 4 to 1 and then gets buffed by a might increasing effect, does it need to gain at least 5 might in order to gain one might?

No, the unit does not need to gain 5 might to increase its total might above 1.

According to the RiftJudge FAQ #450 and the principles of the layering system, Leona, Zealot’s ability is a continuous passive effect that sets a floor of 1 might for stunned enemy units.

When a unit currently at 1 might (due to Leona's reduction) receives a might-increasing effect (a buff), the calculation works as follows:

1. Application: You apply the might buff to the unit's current might value.
2. Continuous Effect: Because Leona's effect is a passive, continuous ability, it reapplies after the buff is calculated.
3. Result: If the unit receives a +1 might buff, its might would calculate as 2. Because 2 is greater than the 1-might minimum imposed by Leona, the unit will indeed end up with 2 might.

Leona's effect does not require the unit to overcome the entire "-8" penalty before it can see any increase; it simply acts as a floor of 1. Any increase that pushes the total above 1 will be reflected in the final value.