So I noticed there is kind of system in the Merovingian dynasty's names. Supposedly it goes Richomer -> Theudomer -> Merovech -> Childerik -> Chlodovech -> Theuderik / Childebert / Chlodomer / Chlotarius -> ...
It seems like most of the time at least one part of the names goes on to the next generation, with Merovech -> Childerik and some of Chlodovech's children being exceptions. The first three all share mer, then come Childerik and Chlodovech, but then Chlodomer and Chlotarius again inherit Chlodovech's chlot.
At the same time I don't see much of a system in here. So was there any system or was it just a coïncidence? Or is it something in between? Is it something just this dynasty did or was it a practice shared by all / other Germanic tribes?
Many Germanic tribes, the Franks (of whom the Merovingians were a ruling dynasty) among them, used two-part names where both parts were meaningful roots in the language. (So, for example, Childeric is Frankish for "fight-powerful.")
Both parts were usually drawn from a relatively restricted (albeit not tiny) set, with some used far more often than others - just as names are in modern times, but with two halves you can mix and match, rather than as individual units. Some being shared from one generation to the next some of the time is a thing that can happen by chance.
I'm not aware of any custom of deliberately passing on an element through a family.
EDIT to add: Frankish name construction is discussed in The Means of Naming by Stephen Wilson.
If you are/speak German I would highly recommend this archive. It's a daily spot on SWR1 where one of Germanys top experts on Slavic/Germanic names, Prof. Jürgen Udolph, talks the listener through the history of their names. He often refers to various naming systems used at different times.