The only centrally located hotel that's an American chain is the Westin Chosun. The Grand Hyatt is technically central on Namsan, but it's inconvenient to reach because of it's mountain location.

Lotte World is better for small kids. It's very compact and is both inside and outside if it's cold/rainy. You'd just take Line 2 to Jamsil Station. Lotte World is the best for little, little kids because of the "Disney-esque" factor. Cute photo ops, their own characters, etc.

You could also go to Seoul Land which is located inside Seoul Grand Park along with the Seoul Zoo (lots of attractions in one place).

Everland is the furthest away and takes longer to get to. For you, as a tourist, you'd have to take the Everland shuttle bus which is like 12$. There are public buses and a subway that are cheaper, but not as simple for you as a tourist. Everland also skewers a bit older. It's much more of a process.

Let me know what other questions you have, I'm happy to answer any and all!