The Book-of-the-Month Club is an organization that (as I understand it) mails you a popular book each month. An "alternate" is someone who wins a contest if the real winner refuses to accept their award (or drops out of the competition or gets sick or dies or something). My guess is that the writer's book will be the "book of the month" if the actual winner refuses to participate. This does sometimes happen: the similar Oprah's Book Club was scandalized when Jonathan Franzen refused to allow his 2001 novel The Corrections to be included.
Another possible meaning of "alternate" here: there could be a #1 book of the month that members automatically receive, as well as "alternate" books that they choose to receive if they don't like the sound of the #1 book. This is basically the same as above, the "alternate" is the not-quite-winner.
Keep in mind this is the noun AL-ter-nate, not the verb al-ter-NATE.
Note also that the author of a book selected as a Book-of-the-Month Club alternate could expect to earn quite a lot of money from royalties. Not as much as the first choice but enough to live on for several years.