In mathematics, what distinguishes an exceptional undergraduate thesis for the rest? How do original contributions, accessibility of the topic (ie more or less prerequisites), novelty of the topic, etc factor in. Would it, for example, be better for an undergraduate to give an overview of well trotted areas or venture into less significant areas and try to pick some low haning fruit?
For me: significant originality, even if that originality is expressed within the context set by the supervisor.
One cannot expect the student to understand what problems are interesting to the community, and which are feasibly solved by an undergraduate. But once the context is set by the supervisor, there can be significant room for originality within that context.
I was working on "locked polygonal chains" in 2D and 3D, and posed the question to my student of whether or not such chains could lock in 4D. The answer was provided in her undergraduate thesis, and subsequently published in a journal: No.
The proof was very much an equal-footing collaboration, but she was inventive and precise throughout. Especially on the algorithm illustrated above, she was ahead of me.
Roxana Cocan and J. O'Rourke. "Polygonal Chains Cannot Lock in 4D." Comput. Geom. Theory Appl., 20 (2001) 105-129.