Kate is a lovely name. It would totally honor your mum, and I like it as a first name. But if your name is short (like Smith, Watt, ect.) don't, or if it is a last name you hear a lot (like Baker, and Smith again). It goes well with long or unusual ln (like Ross-Weatherwurst or Sciaraffa.) Give the child a really unusual or long mn, though (like
Coriander,
Juniper,
Dallas,
Ireland, Periwinkle-not Periwinkle-Jessique,
Thalia, etc.) That way she can always go with that name if it's her type and
Kate isn't. And the cousin: don't worry. I don't think she will have a problem with it, she might even feel like you're flattering her.
Hope I was helpful!