Eventually I think, if they are really serious about an overnight train on the NEC extending into Virginia or beyond, they would be better off running an honest to goodness full fledged LD train. Maybe they could rejigger the schedule of the existing LD Palmetto and have it originate.terminate at Boston, and at the other end extend it to JAX and have it carry Sleepers, Diner etc.. But of course more equipment will be needed for such.
I think you've got two options here:
(1) Run the NEC portion as a day train. For this, you'd extend the Meteor to BOS (this was done for a bit back in the 70s). Given the scheduled times (IIRC the Meteor hits NYP at about 1115 and leaves at 1515) the equipment sits overnight regardless, so hitting BOS at 1515 and departing at 1115 (allowing four hours) wouldn't actually "waste" a set. The bigger issue, IIRC, is the yard situation in Boston plus the question of spare equipment. There's also the issue of "blocking" space all the way from WAS to BOS, especially given frequency limits in eastern CT. A variant of this might run over the Inland Route (which I think is mostly state/agency/Amtrak-owned, and picking up any stray portions might be feasible) - going with the 5.5-ish hours for the old Inland Route trains, that'd put arrival into BOS around 1645 or so (there's probably some time savings from track upgrades on the NHV-SPG section, and some stops could be skipped) and a departure around 0945 (ditto). Again, the question becomes one of capacity controls - but I'd like to think that Amtrak could partially handle this manually to avoid blocking out through traffic, and I'd hope that 13-14 hours was enough time to comfortably turn a train. I'd hope.
(2) If you extend the Palmetto, you'd probably want to rejigger the schedule. Probably a "better" option would be to extend the Shoreliner somewhere further south (I might suggest CLT as a southern terminus, and you can pick which line you run it on). Honestly, extending the Shoreliner to CLT and having folks transfer at WAS (or doing through cars) wouldn't be absurd, and at that point you're looking at needing...three sleepers, maybe?