2010: Space Odyssey Two is a sequel to the MOVIE 2001: A Space Odyssey. It can serve as a sequel to the book, but it is officially a sequel to the movie. 2001 the movie is based near a different planet than 2001 the book, and the second book using the planet in the movie as its starting point. Regardless of this change, 2010 is a fantastic book and another must read as far as I'm concerned.
The character I mentioned that turned out not to be a major player IS a major player in the second book. I was happy to see this. The book also discusses the troubles families can have in long distance/long time relationships. It shows that not all aspects of a story have happy endings. There's also a movie based on this book, released in 1984. I haven't seen the movie yet, but am adding it to me "watch in the future" list.
If you have a science fiction fan in the family, this book is a great option for a gift if you're struggling to find a gift.