There is an exercise in these notes of Pete L. Clark that says that for every $N$ there exist a "nice" curve over $F_q$ that has more than $N$ points over $F_q$(q is fixed). Of course one can prove this using the Bertini type results of Gabber and Poonen but I think those results should be an overkill.
The hint for the exercise says that use a hyperelliptic curve defined by a polynomial with many rational points but I don't understand how that might help because the hyperelliptic curves over $F_q$ can't have more than $2q$ rational point. Is there an elementary proof for this exercise?