Before you post
Start with research
Chances are, if your suggestion is a big change, it or something similar has been requested before. Search carefully for your suggestion on Meta Stack Overflow. If you really want to be thorough, check Meta Stack Exchange too.
See what arguments have already been given for and against your suggestion, if any. Most of the time, these suggestions have also been thoroughly gone over, point by point.
Once you've compiled this research:
Gather data
You want to make any sort of large change? You need to back up your claims, show us what is wrong and why it is worth dev time or community time to try to change it. Even if your suggestion is for a small change, you want to have at least a couple examples of what you're talking about to help illustrate your point.
If at all possible, get stats. This could be from the Data Explorer, or by posting a request for stats here on Meta.
The more you can show us to support what you say and why you feel things should change will help your case. Prove to us that, even if we don't like your specific suggestion, something has to change.
The post
Present your findings clearly and uniformly
Pretend you're presenting information to someone in authority that you respect — your boss, your teacher, your parents. Pretend you have to have their approval to do whatever you're trying to persuade us of.
With that in mind, keep your tone neutral. Stick to the facts. Present only the data, not your emotions. Try to be as even as you possibly can.
If your post reads like a rant, people aren't going to take your want for a constructive discussion seriously.
Clearly explain and justify your suggestion
What, exactly, is it that you want changed? Why will it benefit the site? What, exactly, is wrong that is inspiring this change?
Be sure to be as clear as you can on these points. This will not only help others to see/suggest changes to your proposal, but it will help prevent closure for your post being unclear. We need to know what is wrong, what you propose to fix it, and why your fix would help.
Remember, you're the one that sees the problem and has a solution. Sell us on it.
Be open to feedback
Once you've posted, remember that your idea will be thoroughly picked apart and torn to shreds. No matter how well documented and researched, your post will be gone over with a fine-toothed comb. People will suggest problems and flaws with your suggestion. They will suggest problems with your research and your data.
Don't get angry and fight those suggestions. Take them to heart. Present them with data that negates their concerns, or suggest another way to handle things that will fix what they're concerned about.
Be open to alternate suggestions
If you're correct that something is wrong and needs to be changed, be prepared for answers to be submitted that suggest alternatives to your suggestion. They are no less correct than you are, and it's better to keep the discussion together than to segment it out over multiple posts. This is completely normal behavior.
Don't get into arguments in the comments
If someone is getting heated, stop communicating with them. It's better for your sanity, and you don't want to get carried away and off the point of your post. It'll be a lot easier for you to respond to feedback if you aren't getting heated as well.
After the post
If your post ends up closed/deleted
This does not mean that your opinion is being silenced, that the community does not want your input, or that we disagree with anything that challenges our way of doing things. All it means is that your post did not conform with the guidelines above and in the help center, and that your post did not really add anything new to previous discussions.
We do not close or delete posts just because we disagree with them. We close and delete posts that do not conform to our quality guidelines.
If your post ends up closed as a duplicate
It is possible that despite all your research, your question could be closed as a duplicate of a previous discussion or feature request.
If you still think your question/discussion/feature-request is sufficiently different, you should follow the guidance in "This question may already have an answer here" - but it does not. Read the other question and make sure your question clearly explains why it is different, or why your idea is different enough to warrant more discussion
If you agree that the ideas in your question are the same or similar enough, you could leave an answer on the duplicate instead. Before you do, make sure none of the existing answers already cover the same ground. If you think your ideas are new, then you can leave an answer expressing your support for the idea, explain why it is a bad idea, or propose an alternative solution to the problem. The goal of your answer is to add new information to the previous question to explain why you support the idea and how it will benefit the site. It could help breathe new life into an old idea.
If your post is downvoted
Note that downvotes are different on Meta, and do not always mean that your suggestion is poorly written or off-topic. Especially with feature-request posts, downvotes can be used to show disagreement with a suggestion or idea, in addition to the usual reasons.
So, if you did all of the above, but your post is still heavily downvoted, this may simply mean that you're not quite on the same page as the rest of the community. Try to understand the arguments against your suggestion or idea, and use this to improve your understanding of the site, or change your perspective, which will allow you to contribute more effectively on Meta as well as the main site in future.
Also, don't worry — unlike on the main site, votes don't affect your reputation.
Your suggestion still might not be implemented
Even if your suggestion is very well-received, solves a big problem, and the community really seems to like it, it might not be implemented. The Stack Overflow team still has final say in what gets implemented and what they spend developer time on.
If they do decide to implement it, the implementation could take 6-8 weeks.
Conclusion
So long as you are calm, your post is well explained and supported, and you respond well to feedback, your suggestion should lead to a constructive discussion.