David, I don't know if you are still interested in this, it's been over a year. I just stumbled upon your question in the depths of MO. I often found that Weil restrictions of elliptic curves give nice families of examples on which you can test things.

E.g. take a family of elliptic curves $y^2=x^3+t$ and Weil restrict it from ${\mathbb Q}(i)$ to ${\mathbb Q}$. Writing $x=x_1+x_2i$ and similarly for $y$ and $t$, expanding the equation and breaking it into real and imaginary parts, you get a family of 2-dimensional abelian varieties over ${\mathbb Q}(t_1,t_2)$ given by two equations in a 4-dimensional space,
$$
y_1^2-y_2^2 = x_1^3 - 3x_1 x_2^2 + t_1, \qquad
2y_1y_2 = x_1^2x_2 - 3x_2^3 + t_2.
$$
Alternatively, you fix the elliptic curve, but you let the extension vary with $t$ (e.g. ${\mathbb Q}(t^{1/3})$), or both, and you also get interesting families.

The really nice thing is that as opposed to Jacobians, Weil restrictions are trivial to write down in terms of equations. Over the algebraic closure they are ~~isogenous~~ isomorphic to products of elliptic curves (making them boring), but for arithmetic applications they are interesting. There is a small extension of this construction, when you do not base change the elliptic curve but you "tensor it with a ${\mathbb Z}-$module with a Galois action", which is not necessarily a permutation module. This is explained in Milne's paper "On the arithmetic of abelian varieties" (Invent. Math. 1972) section 2, and it is useful if you want to write down non-principally polarised examples.