I looked at Natura Artis Magistra ("nature is art's teacher"), so I think mens artis is the correct translation of "Art's Mind". How to translate "Art's Mind" (with art in the sense above) to Latin?
You're right, mens artis is "the mind of art". Depending on how you want to use it, you might want a different form of mens. If it's a motto or a name, mens artis is perfect. If there's more context, that might change things. For example, "for the mind of art" would be menti artis.
If you want something that sounds different from "men's" but it's a name, you should choose a different word rather than a different form of mens. The different forms mean different things "me", "I", and "my". One option is animus, which is more spirit than mind.
Another option would be anima artis (the soul of art), depending on the shade of meaning you want.
If you want to leverage ancient associations between the mind and the chest or heart, you could go with pectus artis or cor artis. These would sound a bit anachronistic but also more poetic than your other options.
One thing I would stay away from is intellectus artis. Because while intellectus can mean "intellect" (cf. intellectus agens), when used with artis the much more natural sense is "the meaning of art" (since that reading doesn't require a personification of art).