The
Hobbits were certainly patri
linear. But then, any of them, say, being descended from
The Old Took by a female "intermediary" (mother in
Bilbo's case, maternal grandmother in
Frodo's, etc.) certainly would not be hiding this aspect of their ancestry.
The most extensive (only?) writings on what one could term "Laws And Customs Among The Hobbits" is, as far as I know, to be found in Humphrey Carpenter's "Letters", specifically letter 214 to one A.C. Nunn, a draft. It runs to almost eight pages in my paperback.
"... But the government of a 'family', as of the real unit: the 'household', was not a monarchy (except by accident). It was a 'dyarchy', in which
the master and mistress had equal status, if different functions. Either was held to be the proper representative of the other in case of absence (including death). There were no 'dowagers'. If the master died first, his place was taken by his wife, and this included (if he had held this position) the titular headship of a large family or clan. This title thus did not descend to the son, or other heir, while she lived, unless she voluntarily resigned."
I shorten JRRT' comments on an unusual constellation in the Took family, whose male head automatically became the Thain. Take a look at Appendix C "Family
Trees", "Took of Great Smials". The grandson of The Old Took, Fortinbras II, SR 1278-1380, had married one Laila of the Clayhangers in SR 1314. Laila lived until SR 1402 (but was not at the Long-Expected Party due to "weighty" reasons - think
Bombur). So she remained titular head of the Took Clan for 22 years after her husband's death. Their son Ferumbras III was Thain during this time (and until his death in SR 1415), But not head of the clan. He also never married, apparently because no
Hobbit woman was willing to put up with that mother-in-law Laila. That is why upon Ferumbras III's death, both
titles of head of the Took Clan and Thain shifted to
Pippin's father Paladin II, The Old Took' oldest great-grandson (and in a male line of descent).