The sentiments are normal for a 13-year-old, but Holly's slang scholarship is weak. Her mother Valerie is "in her late 30s", according to the author's description, so she would have been in her prime teenage slang-assimilating years about (say) 17 to 22 years ago, or about 1985 to 1990. I'm sure that "I'm cool with that" was current then.
Searching the Google News Archive for {"cool with that"} confirms my guess, yielding "I'm cool with that" from the Fresno Bee on April 8, 1990, and "If he's totally cool with that and not many babies are that's the second test" from the Doylestown Intelligencer on April 17, 1987.
The OED takes us all the way back to 1959, though with a slightly different phrasing (and in more rarefied social circles):
1959 Esquire Nov. 70H in R. L. Gold Jazz Lex. (1964) 66 ‘Do you want to go to the movies?’ ‘It's cool with me (acquiescence).’
And the use of cool "as a general term of approval" is attested all the way back to 1884.
1884 J. A. HARRISON Negro Eng. in Anglia 7 257 Interjections... Dat's cool!
However, it's a false rumor that Chaucer used cool in the same sense:
c1440 Chaucer's L.G.W. (MS. Gg. 4. 27) 258 Thow..thynkist in thyn wit that is ful cole That he nys but a verray propre fole
Maybe from a 13-year-old perspective, it's still cringeworthy for mom to keep using the slang she learned as a teen. But cool is so old that it's even old news that it stays current ("Why cool remains hot", 1/13/2004).
Posted by Mark Liberman at February 4, 2007 02:36 PM