Swimsuit. Public pools are a big part of daily life here. You'll want to visit at least one.
Shoe cleats. For winter visits. Black ice on sidewalks is common and locals wear them regularly.
Eye mask. If you're visiting in summer, the midnight sun can make sleeping difficult without one.
Reusable water bottle. Iceland's tap water is some of the cleanest in the world. Just use the cold tap to avoid the sulfur smell from the hot water system.
Ziploc bags. Tuck your phone, camera, and wallet into resealable bags whenever you are outdoors. Waterfall mist, horizontal rain, sea spray, and sudden downpours can soak everything in your pockets within seconds. This is not just for Seljalandsfoss. It applies to boat tours, beach walks, glacier hikes, and any time the wind picks up. A cheap bag keeps your electronics safe.
Tissues or a small towel. Even with a Ziploc bag, you will need to use your phone for photos. Keep a small pack of tissues or a microfiber cloth in your jacket pocket so you can quickly wipe your screen dry between shots. I saw a lot of people at the waterfalls struggling with wet phones and no way to dry them. Tissues are small, weigh nothing, and solve a real problem.
Battery bank. Cold weather drains phone batteries fast. If you are out chasing Northern Lights, hiking, or on a long drive, your phone can drop from 60% to dead in under an hour. Bring a portable charger and keep it in an inside pocket close to your body so the cold does not drain it too.
Earplugs. If you are staying anywhere central in Reykjavík or Akureyri, the streets stay loud well past midnight, especially on weekends. Earplugs are tiny and save your sleep.
A small set of medical supplies. Bandaids, painkillers, cold medicine, stomach meds. Most of this is not sold in grocery stores. Pharmacies (Apótek) close at night, and you will need it at night. Bring the basics from home.
Contact lens solution. Bring more than you think. This is one we hear about constantly, both from a frequent visitor on Reddit and from people walking into the local store I work at, panicking. Contact lens solution is not over the counter in Iceland. It sits behind the pharmacy counter, pharmacies close early, weekend hours are limited, and even when one is open, stock is inconsistent. Pack a full size bottle, plus a backup, plus your glasses as a fallback. The same applies to specific saline, eye drops, or any niche personal care item you depend on daily.
A wet bag for your swimsuit. Not essential, but very nice to have. Pools are part of daily life, and a soggy swimsuit in your day bag gets old fast.
A waterproof bag cover. Your suitcase or backpack can get soaked sitting on the airport tarmac, on the way to your hotel, or on a hike. Either bring a waterproof bag or pack a cheap rain cover for it.
A phone case with a strap. Easy to drop a phone into a canyon, a river, or the sea. A wrist strap or lanyard case is cheap insurance.
One comfy outfit for around town. Jeans, joggers, whatever you actually relax in. After a few days in full hiking gear you will want to change into something normal for dinner or a walk down Laugavegur. Locals wear jeans out all the time.