Here are some open problems that I have encountered, sorted by subject. I think they are approachable, and I can send further thoughts if you reach out. I expect the list will grow.
One long-standing program in combinatorics has been to understand tilings of a shape with polyominoes. A polyomino is a contiguous shape formed of squares joined edge to edge. One question one can pose about tilings is what are the properties of the metagraph. A metagraph, denoted $\mathcal M(G(n,m), k)$, has as vertices the possible tilings of an $n\times m$ rectangle with $k$-polyominoes. There is an edge between two vertices, that is, between two tilings, if they differ in exactly two $k$-polyominoes. It is very poorly understood. We do not know if $\mathcal M(G(n, n), \frac{n^2}{3})$ is connected when $3$ divides $n$. One way to explore this problem, presented in Jamie Tucker-Foltz's Locked Polyomino Tilings, is to search for "locked polyomino tilings." A tiling is called locked if, when viewed as a vertex of the appropriate metagraph, it has no neighbors. Tucker-Foltz constructs locked tilings with $k = 3$, $k = 4$, and $k = 5$. It is not known if there are locked tilings on finite rectangular graphs for larger values of $k$, though Tucker-Foltz has code to explore generating locked tilings and computing metagraphs.
I worked on trying to find a locked tiling for $k = 6$ by hand at the Auburn 2024 REU with no luck. However, I was probably insufficiently clever, and it seems totally feasible to find one. This is one potential direction. Tucker-Foltz lists other open problems and directions, including tiling a torus or an infinite grid.