A light rapid transit system has a capacity in excess of 20000 passengers per hour per direction.
A $1B investment in public transport can buy you a whole lot more than 1.7k vph. Hell, you could reserve that new lane for buses and transport a magnitude more people than those vehicles carry starting tomorrow.
It would take a lot more than $1B to make mass transit viable for a significant percentage of those who currently drive on the 405. Just putting in a light rail line here or there wouldn't do shit because few people's destinations would be within walking distance of the stations. You'd also need to massively increase bus service to even have a chance at making a significant dent in car traffic. For better or worse, LA's low density means vast swaths of the metro area would be cost prohibitive to connect with mass transit.
The areas being connected by the 405 (West LA & San Fernando Valley) are geographically quite challenging for transportation planners due to the Santa Monica Mountains. LA Metro plans to build a tunnel through the mountains but it's estimated to cost $20B:
A light rail system over Sepulveda Pass would probably require drilling a tunnel. Early estimates are at least $6B: http://www.latimes.com/local/lanow/la-me-ln-sepulveda-pass-t.... It could get built, but I don't think the final price tag is going to put the lane expansion to shame.
Buses can already use the HOV lanes. One of the expansion lanes (the northbound one) was an HOV lane, so now there are HOV lanes in both directions. Who cares, given how few riders buses attract.
A $1B investment in public transport can buy you a whole lot more than 1.7k vph. Hell, you could reserve that new lane for buses and transport a magnitude more people than those vehicles carry starting tomorrow.