I was a developer for five years then drifted into management for 10+ years, and started doing product management in the mobile space about three years ago.
I definitely see value in (re)learning to code. I picked up Android and Rails books last year and built some demo apps and it enabled me to have more informed conversations with developers. This in turn leads to more trust and respect, per Sami.
It also gives you more confidence. There are lots of people in the industry who feel that PMs at every level need to be hands on with coding. I don't agree, especially if you come from an engineering background, but I do feel stronger having gone through a 'refresh'. When developers talk detail, I have a clearer understanding of where they are at. And I have a better feel for the effort to implement features now that I know Android and Rails better.
That said, I am not a coder and I don't see much value becoming one. To be good, you need to be doing it all the time. Your goal as a PM should be to gain familiarity - with languages, processes, and tools your team use.
The big challenge is time. To get back into coding, you have to build an environment from the ground up. (And yes, it helps if you have a side project of some sort to focus on.) I wanted to extract data from our Postgres database we have hosted on Heroku. To do this I started by setting up a virtual Linux machine on my PC. After figuring that out, I had to re-familiarize myself with UNIX and shell scripting. I then had to install Heroku's tools and learn how to navigate that. I then had to study postgres SQL which is a far cry from the Sybase SQL I used way back. I then wanted to check my scripts into source control so I had to learn about Git and Github. There's a lot here. You can't just put in an couple of hours a week.
Sami's suggestion of shadowing a developer is a great place to start, and if you do choose to go further, partnering with one will help alleviate some of the pain. It is rewarding so if you have the time and interest, I'd encourage you to give it a shot.