@freemo From the Christian things I've been to, my impression is that prayers addressed to baby Jesus relate to things like innocence, purity, and so forth - that is, concepts associated with childhood. Prayers for strength and the like would tend to be addressed to Jesus from the Last Supper to the Crucifixion, and prayers for the departed to Jesus in Heaven.
However, a friend of mine started seminary this month, who will probably know more. I'll pass along any additional info she gives me.
@freemo I'm told by my (Protestant) seminarian friend that this may be due to denominational differences, as the infant Jesus is venerated along with his mother Mary in Catholic tradition. Protestant references to "baby Jesus" are likely ironic today, since it was vigorously parodied in the movie Talladega Nights.
@khird Ahh that makes a lot of sense, thank you.
@freemo Side note - why does the word "prayer" mean the words spoken, not the person doing the speaking? Shouldn't it be "prayage" or something like that, so we can reserve "prayer" for the person performing the action?