Thanks all for the comments.
The guy is actually lives near Los Angeles, and I live nearby. We are friends.
He has an appointment in Santa Monica about 10 hours before the red-eye flight and was looking for options that would allow him to get a really good nap before the flight. I suggested the United lounge as an option, and told him I'd ask here about sleep potential there. Apparently the answer is "None".
We all really like the Daily Grill restaurant in the Westin, so have pretty much settled on that for their time pre-flight. Before going to LAX, we plan to go and see the space shuttle Endeavor in the California Science Center at USC.
I haven't flown on United for years, and have never visited the LAX United lounge. Doesn't sound like I'm missing much. Other than that, I pretty much know my way around LAX. I've been here 40 years.