Property Testing for Independent Sets
This problem has indeed been studied.  Goldreich, Goldwasser and Ron studied it in their seminal paper which kicked off graph property testing, and then, Feige, Langberg, and Schechtman also have results on it in their FOCS '02 paper "Graphs with tiny vector chromatic numbers and huge chromatic numbers".

Specifically, [FLS '02] show that one can distinguish between graphs with an independent set of size $\rho n$ from graphs $\epsilon$-far from being so (meaning, at least $\epsilon n^2$ edges need to be removed to create such an independent set) by choosing a random subgraph induced by $s = \tilde{O}(\rho^4/\epsilon^3)$ random vertices in the graph and checking whether the random subgraph has an independent set of size $\rho s$ or not.   ([GGR '98] showed a weaker bound on $s$ of $\tilde{O}(\rho/\epsilon^4)$.)  [FLS '02] also show a lower bound on $s$ of $\Omega(\rho^3/\epsilon^2)$.