While equity firms are now funding all manner of cases, the primary bread and butter is contingency fee personal injury stuff, although some commercial disputes are getting traction. Plaintiffs' side damages work is easier to predict. Plus, it's not so much lawyers' reluctance to put a number on things as it is that the investor firms can buy a portfolio to spread around the risk -- so the risk analysis is a bit different. The opportunity comes mostly from the fact that the client often cannot pay costs, which can be significant and which the lawyers don't necessarily want to bear with uncertain recovery. The downside for the equity firms is that many cases settle and most lawyers (or at least many) believe they cannot ethically give the equity firm any say in settlement authority, but that doesn't seem to prevent the practice. With interest rates so low, I guess there's a really good market for these kind of "investments." It's still quite lawyer driven, though -- many of the equity firms have developed their stable of lawyers they trust.
It's not unlike what the large contingency firms do -- they take a portfolio of cases, some high flyers, some meat and potato insurance matters, etc. to spread the risk around.