I also saw your post on Vinyl Asylum.
I've moved to a headphone based system since my son was born 5 months ago. You can see my main system listed below. My headphone system was basically downsized from my second system. That system was in our spare bedroom which got converted into the baby room.
By now you should have probably realized that there are as many products dedicated to headphones as with speaker-based systems. There are many headphones to choose from ranging from $60 to $5000(?). Grado, AKG, Sennheiser, Sony, Audio Technique, Stax, Beyers, etc. Grados and Sennheisers seemed to be the most popular ones. And did I mentioned there are also phones you stick inside your ears if you prefer.
As for amps, you also have your varieties of solid states & tubes. Most of the headphone amps are actually more like an "integrated" amp in that they have build-in volume control. Some of the Hi-End manufacturers, lik CEC, Lehman and PS Audio, have recently taken interest in headphone amps. So you will see more and more headphone amps becoming available. There are also some preamps with headphone jacks that are quite good.
Like everything else, it comes down to personal taste and system synergy. You will probably get 50 different answers, and I can tell you that non of them will be the right answer for you. You will have to decide which combination suits your personal taste and budget. Besides, you haven't mentioned which cartridge and phonostage you will be using. Yes, you will need a phonostage as all headphone amps require line-level signal. Best thing to do is to attend one of those Head-Fi meets in your area and audition as many combination as you can. I was lucky to have a chance to audition several SS and tube amps prior to purchasing my amp. But again, I purchased my amp based on a review, and I was just lucky that the outcome turns out to be great. Best way to start is to spend some time here and do some reading of people's experiences with various headphones/amps. You can find out if there's a meet in your area as well.
You can read about my headphone search journey in my review of my headphone amp Consonace Cyber 20. Several of my buddies have listened to my new headphone setup and commented on its "LP-like" qualities and amazed at the reproduction of female vocals. I've meant to hook it up to my main rig and listen to some LPs, but I've been so happy with the sound coming out of my CDs (I would never imagine in a million year that I would be saying that) that I could not unhook the amp from the system. Not even to unscrew the plates to take a picture of the inside so I can finalize the review and submit it to Head-Fi.
As for my $60K main rig. It has become my 5-month's old son's personal stereo, playing soundtracks of Totoro, a Japanese animation for kids, and classical music.