You can try making a website/app which provides visualizations of the vairious "textbook" algorithms, I think that would span across algorithms commonly applied in competitive programming problems.

My personal advice is to not strictly look for projects which directly apply the "textbook" algorithms or pick an algorithm and then force a project on it, because often times most of decent scale projects teach you certain practices which can help your competitive programming skill and come with their fair share of algorithms. Also, with the above stratergy you might end up with something artitifical or toy, limited in it's educational value.

It also depends on your interest(games, language theory...) but it might be worth looking into foundational projects such as — interpreter/compilers, tiny operating systems, raytracers, games(without using a game engine), etc. I think these projects are quite involved with theoretical bits, moderate/large in scale, also make you explore lot of things under the hood.