Whose surname will the baby have?
If the baby will have your DH's surname, which is presumably your PILs' surname too, she will already share a name with her MIL. And surnames are much more important than middle names.
So if she's just having one middle name, it should be one from your side of the family.
However, if your DH won't agree to that, what about your DM and MIL's middle names - is there a name in common somewhere?