There's something I'm surprised other answers don't address directly.
Use your content - content is king
You're essentially re-marketing your site to what in relationship marketing might be called 'warm leads', people who have some relationship with your product but are not currently interacting.
So use the product - the thing people come for in the first place. Use the interesting content, interactions and activity of your community.
Don't beg, don't talk about re-designs or non-content site features. This would be like a pizza company trying to entice ex-customers back with their new, sturdier pizza boxes. That may be a valuable improvement to the overall experience, but not it's not the thing people come for.
Use the content that people come to your community for. Use your equivalent of pizza:
- Popular posts or discussions. Suppose you run a forum about drones. There's a popular discussion about some new drone on the market. Your ex-users probably have an opinion on it. So pick a discussion-worthy post or comment and use that as the draw. "JohnK says, 'The [whatever] is the best quad drone ever'. Do you agree?". The kind of thing people come to your community for - the equivalent of this month's top selling pizza.
- If you have the tech, relevant posts or discussions based on an algorithm and your data on your ex-users interests. Send JohnK's discussion-worthy comment about quads to ex-users who participated in discussions on quads. The equivalent of your ex-customer's favourite pizza.
- Save features updates for if you know that they might be the reason people might have drifted away: fixing complained-about bugs, offering much-requested services - things that might have impaired feeling able to access the content. The equivalent of "We now offer home delivery!" or "Slow no more: We now deliver in 20 minutes, or the pizza's free!" - not "We have new, branded delivery bikes!".
Some examples of major sites who use this approach:
- Twitter send "Popular in your network" emails, with tweets that their algorithim thinks lapsed users might find enticing. They send me posts they know these people like me want to interact with, so they have every reason to think that I'll see them and be motivated to interact
- Quora's "Quora digest" emails have as their subject line the most popular, enticing question. They know it's a question people like to click on, questions are what people come to their community for, so if anything makes me come back, it'll be a particularly tasty-looking question.
It's better to prompt action than persuade someone to make a conscious decision. People dither and procrastinate before acting on decisions. People rationalise and justify their actions with new attitudes after they act. There's lots of research in psychology that shows changes in attitude follow changes in action much more than changes in action follow changes in attitude.
If someone's response is, "No way is the [whatever] the best quad drone ever", and immediately they're posting in your community again, voicing their opinion, interacting, that's a better result than "Huh, maybe I'll give that site another try. But first I fancy a coffee".