Basic Household Tool List

(Edited to add suggestions from the comments and the general philosophy of this list)

• Tool bag (at least 15 inches long)
• Hammer (12 or 16 oz)
• Rubber mallet (white head preferred, non-marking)
• 6 inch Needlenose Pliers
• 8 inch Slip Joint Pliers
• 8 inch Crescent/adjustable wrench
• 10 inch Slip groove (large jaw) pliers
• Box cutter (optionally folding) + set of replacement blades
• Hacksaw + spare blades
• Scissors
• 25' Tape Measure
• Set of precision (small) screwdrivers
• #1 and #2 Phillips head screwdrivers
• Small and large straight blade screwdrivers
• Multi screwdriver with torx/phillips/straight blade heads
• half-round file (h/t @patience_limited)
• a metal putty knife or a 5-in-1 combination paint scraper (h/t @patience_limited)
• roll of twine
• roll of blue masking tape
• roll of Gorilla tape
• roll of black electrical tape
• super glue
• small bottle of wood glue
• 2" paint brush
• torpedo level
• small flashlight
• a strong neodynium magnet - to use as a stud finder and for many other things (h/t @the_man)
• 9v batteries
• AA batteries
• AAA batteries
• 50 1" #6 wood screws (zinc/steel, not brass)
• 50 2" #8 wood screws (zinc/steel, not brass)
• small box of 1.5" finishing nails
• light and heavy duty drywall anchors
• course and fine foam sanding blocks (h/t @patience_limited)
• 5 gallon bucket
• safety glasses
• N95 masks
• leather work gloves
• foam ear plus (h/t @the_man)

Bonus list (things that are more expensive or nice-to-have upgrades):

• a stud finder - the cheap ones are worthless, buy a good one or not at all (h/t @DeaconBlue)
• a battery powered drill + bit set (h/t @patience_limited)
• a set of metric and imperial allen keys (h/t @sublime_aenima)
• a basic socket set with a mix of metric and imperial sockets (h/t @auk)

The philosophy of this list:

A good starter set should include tools and supplies someone will be glad they had that they didn't know they needed. Even if they aren't "handy", a handier friend might use them (and they learn something in the process). It doesn't include very specialized things like demolition, electrical, or plumbing tools. If you're going to undertake projects in those areas, you should know a little about what you're doing, and you'll be able to buy the right tools for the specific project. As as electrical engineer, I particularly don't include a multimeter because I don't want to encourage someone to mess around with electricity if they don't understand how it might hurt them.

This list specifically does not suggest brands (except where the brand is the thing, like Gorilla tape). Opinions on what brand is best vary widely, but my general suggestion is this. Don't break the bank buying high end tools for someone who may never use them. A $2 hammer is going to be fine for someone who uses it once a year. If they are using them often enough to want something better, they can buy something better (or you can buy it for them).