While it’s a good idea at heart, I think the teams are so dis-harmonized these days, getting the regulations in place would be hard. If I remember correctly, the top 4 constructors were not allowed to run a car in those sessions 10//9 years ago. This massively helped BAR in scoring 2nd in 2004, as well as McLaren in 2005 to score the same result. I could imagine Ferrari and Red Bull being quite pissed in the press about this, had Mercedes been allowed to get extra mileage every race this season.
The easy way to avoid this, would be to allow all teams to have a 3rd driver on Fridays, but a 3rd car, as between 2004 and 2007 would probably be far too costly. In that, you’re risking a lot at tracks like Monaco, Singapore and Melbourne. I think the number of drivers who actually get their hands at the wheel in those sessions would disappoint, especially for teams like Ferrari and Red Bull.
As far as costs go, apart from extra spares, I think it can be done. Have a one hour session or so before FP1, drivers with less than 2 GPs experience in the past 2/3 years (which would allow a driver like Gene to drive, but someone like d’Ambrosio as well). Pirelli could spice things up a bit by offering teams an extra set of tyres for a driver from GP2, FR3.5 or similar to drive. I would hope GP2 would loosen up about their drivers getting a shot in an F1 car in that case, though.
Mondays would be less practical, I think. A lot of staff would already have to return to the factory to work there, teams could have already found out on Friday that certain new parts don’t work and head home regardless.