Are you asking: (i) whether you have enforceable legal rights in relation to the idea; or (ii) whether there are any legal issues associated with publicly claiming ownership of an idea in these circumstances?
(i) Enforceable legal rights?
The types of idea that you can express in a tweet will not usually be protected by intellectual property rights or other legal rights (although there are some exceptions).
(ii) Risks of claiming ownership?
The risks depend upon the situation. For instance, if the claim amounts to an allegation that someone else is a liar, there could be defamation issues.