Please note that not everything in the official Prima Guide is accurate, however, according to the official guide it says:
Ask to Stay Over is under the Friendly Commodity (So make sure in the upper left corner of the screen the two Sims are under Friendly for this social interaction to show up) and the actor ages are: C, T, Y, A, E
I think I remember having a child and seeing this show up and was surprised because I didn't know children could ask to stay over...or didn't remember because there's a lot in TS3 to keep tabs on, lol.
Also note there is a specific time range when this social interaction shows up such as 5pm - ?. May be inaccurate since I do not recall writing it down and investigating this further as I did with the times and days of breakfast, brunch, lunch and dinner.
Edit: To clarify, the Ask to Stay Over social interaction is both under the Friendly and Amorous commodities under the appropriate actor ages.