Answer: C

Small Hint:

Area ratios are affine-invariant, so choose convenient coordinates for triangle $ABC$

Big Hint:

Locate $D,E,F$ by the one-third conditions, intersect the three cevians, and compare the two areas with determinants

Solution:

Use $A=(0,1),$ $B=(0,0),$ $C=(1,0).$ Then $$\begin{aligned} D&=\left(\frac23,0\right),\\ E&=\left(\frac13,\frac23\right),\\ F&=\left(0,\frac13\right). \end{aligned}$$ Intersecting $AD,BE,CF$ in pairs gives $$\begin{aligned} N_1&=\left(\frac47,\frac17\right),\\ N_2&=\left(\frac27,\frac47\right),\\ N_3&=\left(\frac17,\frac27\right). \end{aligned}$$ The determinant area formula gives $$\begin{aligned} [N_1N_2N_3]&=\frac1{14},\\ [ABC]&=\frac12. \end{aligned}$$ Hence $\frac{[N_1N_2N_3]}{[ABC]}=\frac{1}{7}.$

Thus, the correct answer is C.