As Rafael has already commented, we would typically put cor in the ablative case here, and say nobilis corde.
Cor, like English “heart,” is not just the organ (or, as Lewis & Short put it in their inimitable style, “the chief source of the circulation of the blood”), but also the soul, feeling, or courage. I think it fits quite well, and there is no need to look for alternatives, although animus would also be a good fit.
Nobilis can indeed mean “noble” in the idealistic sense (not as in “aristocratic”), as the Valerius Maximus example shows; however, it is quite tightly associated with the idea of high birth. Perhaps you might prefer the more general term generosus.
Who or what is noble of heart? Presumably your family, that is to say, your familia, gens, stirps, domus. These all happen to be feminine, so I would suggest generosa corde. (In case you are wondering, in principle the same issue arises with nobilis, but that form can be masculine or feminine – neutral would be nobile.)
It is perhaps worth mentioning that nobilis cordis would not be incorrect either. This is because nobilis is attested with the genetive, as in Curtius et animi et generis nobilissimus adulescens (Valerius Maximus 5, 6, 2) However, some editions (including the one linked here!) have nobilissimi instead, which may perhaps be due to an uneasiness on part of the editors with the genetive construction. In that reading, nobilissimi modifies et animi et generis, and the meaning is “of the most noble mind and pedigree.” This would also work with nobilis cordis (alternatively generosi cordis), but I don't think it sounds so good as a standalone motto.
