What we stash depends on the kids' ages.
We keep looseleaf paper (college ruled when they are older), pens (black, blue, red), colored pencils, #2 pencils, rulers, scissors, glue, tape, folders, spiral notebooks, 3 ring binders, book covers, pencil sharpeners...anything that is unused the year before goes back in the bin.
For little kids, I'd include crayons, glue sticks, construction paper, markers and a pencil box. The class will let you know if they need sanitizer, wipes, paper towels, etc.
I'd also get labels (like skinny minis from Mabel's Labels) for things that go to the classroom and might get lost, like scissors.
For gifts, I keep small toys in the age range of the kids' friends for small kids. Older kids...eh, it's harder. Maybe gift cards that don't expire. Coloring books, reading books, craft items, small lego sets, and gift bags and colored tissue and cards of all kinds. My mom had a suitcase we "shopped" from as kids and I liked that idea.