What is the origin of the phrase "Put one in the wood for him"?

Solution 1:

"Put one in the wood" is a competitive dart throwing term. The bullseye is made of wood while the outer board target regions are made of other material such as horsehair.

Solution 2:

It's usually put one in the barrel...

I asked the landlord for a pint, pointing at Duncan. He pointed at two full ones lined up at Duncan's elbow but agreed to put one in the barrel for him.

Admittedly it's a slightly odd usage, in that the pint of beer is already in the barrel, so reserve might be a better verb than put. I'm from Sussex, and I note OP is in Kent, so perhaps the pint / wood / barrel form is essentially a South East UK usage (but it's probably more widespread).

I don't think there's any reason to look to an "origin" as such. If your friend arrives at the pub after you, you might say "There's a pint in the barrel" (already paid for by you). Americans don't drink pints, but I'm sure if one was eating a slice of pie in a diner he could say "There's a slice behind the counter [for you]" when his friend arrived, and expect to be understood.

Here's a link to sociolinguist Kate Fox in Watching the English (2005) talking about it.