I've got a high-powered VC on my BoD but he seems checked out during our meetings. Is there any way to re-engage him in our governance?

How are you doing as a company? From my experience, board directors check in and out based on how well you are doing and how busy they are. For starters, they are always busy! Knowing that, they tend to pay attention if you fit into one of three categories:

1)You are their "pet" project and they want to make you succeed for reasons outside of the typical ROI for their firm.
2)You are killing it! (and congrats if that's the case). VCs love to follow their wins closely b/c it gives them a great story to tell their partners and LPs
3)You are getting killed! (sorry about that). If you are getting burned you become an embarassment for them. There's no hiding those reported numbers every quarter and they are now trying to figure out how to fix you, marry you off, or kill you. Sometimes you pass a point of no return in these cases and you will not get them back unless you move to #2.

If you are in area 1, 2, or 3 I would bet you'd be getting attention, so let's not address those now. If you are outside of these areas it can be very hard to get their attention. Remember that they have 8-12 other active investments and are getting pitched daily. If you are just humming along nice and quiet-like, you will find that you're not a "P1" for them and they will be hard to get attention from.

Regardless of what they think of you, you should be well within your rights of asking them to re-engage. See what they say when you casually mention, "hey, we could really use some more of your time on the board." If that doesn't do it for you and you continue getting nothing, you are just wasting a seat and wasting all sorts of efforts they could be bestowing on you.

If so you can, and i have seen this work, ask for another partner at the firm. Note that this will really wake up that partner if they are at all "wakable" (new word). This can either really piss them off, or it can wake them up and get them working for you again, or it may bring them to the realization that they are screwing you up but they are too busy, so you may well get that new partner.

Other things to try:
-Do you know other partners at the firm? Can you "back channel" to them and see what's going on with your board member? You'll want to know them pretty well before you ask this question.
-Do you know other investors who know that person? Can you back channel through those folks? Like above, you'll want to know this investor well.