The toolbox is a good idea but also putting together a care package of sorts with all the incidentals one needs that add up quickly.
A plastic storage tub filled with paper towels, TP, napkins, cleaning supplies, sponges, scrubbies, trash liners/bags, soaps, detergents, bleach, dusters, etc.
I did this for my daughter when she moved to her first apartment and after she unpacked everything she had the storage container to use for other things to, turned into a laundry basket in her closet. haha

a set of towels

Every home should have a fire extinguisher, so one for the kitchen and one for the garage would be a thoughtful, lifesaving gift. Although it’s not a ooooh ahhhh gift.

A really good knife. I learned that one from the new kitchen thread a few weeks ago.

A toolbox is essential, and so few young people realize this when they get their first apartment. Now, if only people could think to give these to their (grand)daughters as well as sons, that would be even better.

A drill or a ladder would also be good ideas.

Glassware is cool

A plant.

A small tool set. Or a finishing hammer engraved with their name and the date.

First aid kit. You can buy something prepackaged.

Basic sewing box. Very few of us will turn out to be tailors; every one of us will need to reattach the occasional shirt button or hem.

Iron, ironing board, and a padded cover for the ironing board. Everybody needs to press a shirt or refresh the crease in a pair of trousers.

Really basic kitchen goods. Not the fun stuff, such as food processors and pasta machines, but measuring cups and spoons, mixing bowls, etc.

A step ladder. Even if you’re really tall, there are times that you need a step ladder.

An easy to care for plant.

Easy to care for = artificial!

Great ideas, all. A vacuum sweeper would be good, too, even if it was a used one that ran well.

Along with @ibstubro ‘s suggestion, a used Compact vacuum is a winner!

I vote for a high-end 7” or 8” chef’s knife and (if you can afford it) a 4” multi-purpose paring knife.

If your budget is limited, then I love the idea of a tool chest.

@gailcalled, I was given a fully stocked Craftsman tool box and tools, as my grandmother was a Sears retiree.

On the other hand, I’m still using the Ecko 8” chef’s knife I bought at Dollar General about 1985. I bought 2 because I had to provide my own knife to be chef cook at a full service restaurant and I liked the first so much, I bought a back-up. I still have them both.

A potted plant. It gives off oxygen.

@Jonesn4burgers said: I think a laundry gift pack is a nice idea. A laundry basket, with some neccessary items, perhaps a bundle of wash cloths, and definately a roll of quarters, if they don’t have machines in their apartment. When moving, it is easy to forget to have the quarters on hand. It can be put together quite cleverly, and everyone has to do laundry

A pedigree cat and a scratch pole.

@Hypocrisy_Central LMAO!
That’s a gift that may not be well received, might as well give them a 3 foot Iguana and a 400 gallon aquarium or a Parrot and a 5 foot cage.

Everyone has given wonderful suggestions.I once bought a Tree-to-be kit as a housewarming gift for my cousin. She was into gardening and she absolutely loved it. She planted it with her kids and I am sure it would be fun watching it grow over years.

The boring fundamentals;

Broom
Dust pan and brush
Mop and bucket

I use my ancient ones almost daily.

I’d go with kitchenware or tools. They’re not the most flashy gifts, but they will definitely get the most use.

Bed linen.

I just bought a friend who recently moved into a new place a set of margarita glasses, a bottle of my favorite organic margarita mix, bottle of tequila and a Margarita scent Yankee candle. She loved it!

This is something I put together for myself when I moved into a new abode. It really made everything easier and became a great gift for other’s moving presents! Plus it’s economical enough that one does not have to do without food for several days in order to afford it!
I bought an accordion file, a three ring binder with tabbed dividers that flipped over to provide a pocket for the section it headed. Then picked colored pens, blank tab tags, and lined paper of different colors.
The accordion file sections were for the set up of the first paper contracts, $ deposit dates, leases, utility agreements, morgage materials/leasing agreements, mover contracts, insurance info, any additional company info etc.
The tabbed notebook held the warrantees of new appliances, schedules for repairs upon signing, sections to keep monthly utility useage information, listings of areas needing attention, or recorded problems for the rental management/repair.
I did add a calendar to easily help keep track of start and stop dates, etc. and items that I didn’t want to keep on my phone.
Any technical failures or losses, you still have what you need in a hurry.
If you want to ‘go designer’ there are catalog companies who have patterned files and binders that match, labels, etc. You could even order printed address labels for their new nest!
Woo-Boy! Suzy Homemaker watch out! I’m on a roll….yow! :]