Knapsack and set cover-like problem
Your problem is an instance of the following problem:

VECTOR-MINIMIZATION:
Input: vectors $v,w \in \mathbb{R}^m$, a threshold $\gamma \in \mathbb{R}$
Goal: minimize $v \cdot x$, subject to $w \cdot x \ge \gamma$ and $x \in \{0,1\}^m$

In your problem, $v=(v(t_1),\dots,v(t_m))$ and $w=(w_1,\dots,w_m)$ where $w_i = \sum_{t_i \in r_j} 1/|r_j|$, where the sum is over all $j$ such that the set $r_j$ contains the element $t_i$.

So, your problem is an instance of VECTOR-MINIMIZATION.  In turn, VECTOR-MINIMIZATION is basically an instance of the knapsack problem (it is at least as hard as knapsack, and thus NP-hard).

That raises the question, is your problem any easier than the general VECTOR-MINIMIZATION problem (or knapsack)?  In your problem, each $w_i$ has a special form: it is a sum of a polynomial number of reciprocals of polynomial-size integers.  I don't think there's any way to use the special structure of the $w_i$'s (and there are exponentially many possible values for the $w_i$'s), so I think your problem is NP-hard, but I don't have a proof at the moment.

That said, you can still use the standard algorithms for knapsack-like problems: e.g., integer linear programming (ILP).