Uncle passed away, sons not frum

Hello,

It is good that you are caring for your uncle’s neshama. You and your sons can learn Mishnayos and say the tefilla, “ana Hashem…” afterwards. Your father can say kaddish when he is davening with a minyan. If you want, you can pay someone to say kaddish, but you are not required to do this.

As far as naming your children after him, sadly, the fact that no one will be named after your uncle is not something you need to consider. From what you wrote, it seems your cousins married out, so their children are not considered Jewish. One reason to name a child after someone is to connect the child’s neshama to the niftar, which is why we often name after tzadikim. If your uncle did not live the way you hope your children will, you would not want to give them that name or its spiritual influence (see Rashi Yuma 38b D:H d’lo maski, Maharsha Taanis 28a D:H Hu omer, Shut Chasam Sofer 2 Even Haezer 25). We are not here to judge, and we do not know the challenges the older generation faced. Still, since he and his children are not religious, you would not want your children or grandchildren to follow that path, God forbid.

Best wishes