The only thing about Hyde Park is, it's surrounded by pretty high crime neighborhoods. considering, you are moving in with kids.
I live north west, and the only great thing is the space, lot's of back yard, even for the two and three flats. If I didn't love and require my back yard, I wish I lived in Rogers Park. It is diverse, it's the only neighborhood with Waldorf school (in case you want to take some parent-child classes) and it's by the lake... but it's more crowded than here, like most of north side.
But if diversity is your main concern, I wonder if Hyde park is your best bet. See, here in my neighbood it's diverse, lot's of hispanic, european and arabic, not so many african americans though, and Rogers Park has many african refugee families, but as you might guess they kind of keep to themselves. In a university area, you get many like minded smart people of all backgrounds together. But I have never lived there so I should be quiet!!
Lincoln park, lake view, roscoe village and lincoln square are pure white ;) As nice as the neighborhoods are getting, (clean, cute boutiques, cafe's, european feel) I just couldn't live there.
When ever I talk about the dire situation of diversity in chicago (even in diverse neighborhoods, the white, black and hispanic keep to their own company) People seem to have good things to say about Oak Park, the first suburb west of chicago. with good public transport to city. There people of different color are of same economic background, therefore feel equal and mingle! Also these border suburbs are not suburban in a typical american way, oak park is very cute to walk around too, but it is very quiet. Hyde Park will give you that urban feel.