It's an interesting idea, I've kicked something similar around with politically-minded friends for years. I don't know that a completely random group is the answer, but a hybrid approach might solve some critical problems:

- A largely unrecognized problem with our legislature here in the US is vote inflation: the number of representatives in Congress has fallen way behind the population growth, so that one rep is shared by a much larger group of constituents, devaluing the individual vote of each constituent and making it less likely that a given voter has a personal connection with their legislator.

- The increasing partisanship has reduced the number moderate and independent voices in the legislature.

We could increase the number of representatives in Congress by tripling the number of reps from each district, which would bring the rep-to-voter ratio back more in line with where it was when it was essentially frozen in 1929. Then one of those new reps would be chosen at random from a pool. Since the distribution of moderates in the general population is much higher than in Congress, this should have the effect of moderating partisanship.