Actually useful tools are the ones that are (expensive) well made. It's not very fun when the handle breaks; cheap tools can be dangerous. If they are responsible, they may own some of these tools for the rest of their lives. I'd suggest a minimalist approach to mitigate budget concerns.
- They need a hammer that the head won't fly-off of, needle nose and slip-joint pliers, a quality tape measure and screwdrivers; not a bit kit. Though it would be a great addition that could add some random hex, torx, sockets, ect. An adjustable wrench would be nice but not crucial if you have pliers.
A bit kit may actually be the most useful thing to them at this time, for taking toys apart, changing batteries and whatnot; kid stuff. As a child I received a cheap tool box, mostly geared towards woodworking; it was (and would still be) useless to me. Focus on items that will stand the test of time and be useful for the duration.