The housewarming gift has one job: to be used in the new home. The failure mode is buying something decorative they'll feel obligated to display, or something so generic (candles, flowers) that it disappears in the excitement of moving day. These picks are useful, lasting, and don't require you to know their decorating style.
Everyone in a new home needs a toolkit and almost nobody has a good one. OXO's essential home tool kit has a hammer, screwdrivers (multiple sizes), a tape measure, a level, and pliers — all with the soft-grip handles OXO does better than anyone else. It comes in a case. It's $40 and it will be used within the first week. This is the safe bet for any housewarming gift.

New home = new kitchen = a chance to get the pans right. Lodge cast iron is the housewarming gift that gets used for decades. It's $24, not flashy, and exactly what they need. Add a cookbook if you want to spend more. This is the kitchen version of buying someone something truly practical — they'll use it every week and think of the gift every time.

A new home means stocking cleaning supplies from scratch. Grove Collaborative's new member offer gets them a free starter set — Mrs. Meyer's hand soap, dish soap, multi-surface spray, and more — with their first order. You can share the referral link or purchase Grove gift cards. It's a practical gift that sets them up with better cleaning products than they'd otherwise default to.
New kitchens often don't come with cutting boards, and the cheap plastic ones people default to are bacteria traps that don't last. The Boos Block is the cutting board that home cooks upgrade to and never replace. It's end-grain maple or walnut, self-healing when oiled, and beautiful. The 12x9 block is $80 — more than most housewarming budgets, but if you're pooling contributions from multiple people, this is the right answer.
Nobody buys a good trash can for themselves. The Simplehuman 45L semi-round step can ($80) is the one that every person who receives it immediately understands — the lid closes quietly, the liner pocket holds bags so they never fall in, the brushed steel doesn't show fingerprints. It's unglamorous and profoundly appreciated. For a new home, it's genuinely the right gift.

The OXO tool set ($40) for first-time homeowners, Lodge cast iron ($24) for anyone who cooks, or a Grove Collaborative starter set for anyone starting fresh on cleaning supplies. All three are immediately useful.
Practical. They have moving chaos, don't know their aesthetic yet, and have a hundred things they need. A practical gift that gets used is more appreciated than something beautiful that has to find a spot.
Space-saving, multi-purpose items: Lodge cast iron (one pan for everything), OXO tool kit, a great cutting board. Avoid furniture, large art, or anything that takes up floor space they don't have.