Owing something to the orignal meaning of the word, in computer-speak this refers to a computer process-- such as the installation of new software-- which the installer cannot or dare not leave, because the installation is not fully automatic and might require input from the installer during the install.
Computer Geek 1: "Lets go grab a Starucks, want to?"
Computer Geek 2: "Can't. I'm installing new software."
Computer Geek 1: "Oh, just let it install itself, automatically."
Computer Geek 2: "No, this software requires some heavy babysitting."
Babysitting (noun): The often humorous misnomer used by parents when they're actually just doing their regular parenting duties. Commonly applied when one parent takes sole charge of their child while the other is out, reinforcing the age-old joke that parenting can sometimes feel like a part-time gig.
Friend: What are you doing this weekend?
Parent: Oh, you know, babysitting. My wife is going out, so it’s just me and the baby gearing up for a wild night of cartoons and cheerios.