Is there a specific night of Chanuka that is better to give out Chanuka Gelt?
The Lubavitcher Rebbes used to give Chanuka Gelt on the fourth or fifth night.
R' Chein explains that there are two reasons why one may want to give on the fifth night:
1. It never falls out on Shabbos, so one can give gelt on a consistent day.
2. On the fifth night there are more candles lit than not lit (which makes it a somewhat auspicious time).
However, some write that the custom is to give it every night, just add a little more on the fourth or fifth night.
My grandmother would give us 1 piece of gelt for every candle lit that night. There also tended to be 1 question asked for each candle lit, including the shamesh.