Two of my DC have two middle names, and the other has one. They all have first names we chose because we liked them.
DC1's middle is my granny's maiden name. She had recently died.
DC2's middles are my father's name and GFIL's name (father of MIL).
DC3's middles are FIL's name and a name from my mother's family, held by my brother, uncle, great-uncle, etc.
We only got into two middle names with DC2. We have offered to add another for DC1, although it will be fiddly and expensive to get a deed poll for a minor. He wants to add DH's name.
So ultimately each of them will have two middle names, one from my side and one from his.
There are lots of family names we haven't used - all the girls' names for a start, but also my grandpa's name which is my ex's name, etc.
I think since they get a his-side surname then it should be ok not to pick a middle name from his side. We all have the same surname so it wouldn't "count".
I do think it would be nicer to have one from their side, though. It can be a very tenuous link, of course! It might not matter to DP but might end up mattering to DC. Could you get a list of all the family names, including grandparents' middles and maiden names, and see if any of them appeals? You might spot an Anne and say you chose Annabel to honour her, for example.