I believe that DLC on XBL uses DRM. Wow, acronym overload, let me try again.
Most (if not all) DownLoadable Content on Xbox Live uses Digital Rights Management. Even if the content is free, I think that this may be true. Not the most intuitive or logical thing, but based on complaints from folks who have this issue, I'm pretty sure that the Optional Media Update is treated like other DLC.
It's their way of ensuring that people who paid for content can use it, but people who didn't pay for it have to show that they did. In regards to (what I think is all, both free and paid) content on a 360, here is how is works.
Gamertag A downloads the update onto Console 1. Later, at a friends house or in his home, he uses Gamertag A to download it onto additional Consoles (2 and 3, etc.).
Console 1 will be able to use the content, with or without Gamertag A signed in. It was the original console to get the download and will have online and offline access to the content. Consoles 2 and 3 however will require Gamertag A to login to XBL (to verify that an authorized account is trying to play the content) in order to use the content.
If Console 1 is replaced (lets say due to a repair) with Console 4, then Gamertag A should use the DRM Migrator tool (mentioned on the xbox.com site) to make Console 4 into the "primary storage location" of all downloaded content. Once that is run, it is like an account recovery and invalidates the "original location stamp" for Console 1 and transfers it to Console 4.
So, I am guessing that you may have DL'ed that content a while back and since have had a swap of your console due to a RROD or other warranty repair or you have multiple consoles? If so, the Migrator Tool may help. If not (and it's possible that the above scenario is totally off base for your situation), another workaround that might resolve the problem without the Migrator Tool would be to find the update on your HDD, delete it, create a silver (free) XBL account, and use it to re-DL the Optional Media Update to the console and hope that since you are using a "new" account to DL it to this console, the console will mark that update as "downloaded to this console first" and allow off-line access to the content.