I am design a page which populates a lightbox on the page with medium size of information on the lightbox itself. I also have to design this page to be responsive on a mobile device particularly smart phones. What's a common practice for this?
May I suggest that when the viewport gets too small you'll simply remove the margins around the lightbox, remove the close button and switch over to what Apple refers to a "full screen modal view"? Pretty much like this:
I wrote about some deliberations that were made over design and usability of responsive lightboxes. The main things to keep in mind are:
Be aware that lightboxes are intrusive! Only use a lightbox if there's no other way to showcase a piece of content. If you must use one, make it as easy as possible for the user to understand that they're still on the same page as before and they haven't done anything drastic.
Users expect that if there's more than one image or video on the page, they should be able to arrow key or swipe or click through to successive content without having to close the lightbox and open it up again to view the next one.
If you're showcasing video, make sure the video's UI and controls aren't obscured or covered by your navigation elements. This also goes for keyboard shortcuts, but sometimes is might be appropriate to override them.
If you can at all help it, don't let your responsive images scale past their natural size. Too large and they'll be blurred, too small and they're not even worth lightboxing.
Decide whether you want to fix the position of the lightbox on the screen or allow it to be scrolled off-screen. There are advantages and disadvantages to each way, and its up to you to decide what's more appropriate for your use case.
Based on the op's explanation in his comment, it appears that the issue isn't about 'lightboxes' per se, but more of a modal window/dialog. The user needs to be presented information, and agree/disagree with it before they can take any further action. A lightbox is one way to handle this, as is an old school modal, or an intermediate page like a user agreement. If the website already uses a lightbox, and it can be resized for mobile to keep the look and feel of the site consistent, it would be the most practical solution.