What is the best way to promote a news aggregator site?
You should probably focus on SEO above all else. 
Of course, the nature of a news aggregator means you will be competing with much bigger and better SEO’d sites.
How do you aggregate the news? Clearly being the “first” to repost an article is likely key.
Thanks for your answer. You’re right, the SEO is an option. Unfortunately, it is not enough to be first, if the site is not promoted in the right way.
I forgot to mention, I aggregate only news headlines into my site.
One of the primary things which you need to focus on is promoting your site through social media marketing. SEO would help you a lot but then since you say that you merely update the news headlines, SEO is not really required. If you go on a big scale and start buying news from agencies, then probably you could switch to SEO.
You’ve got to ask yourself - what are you doing different to convince people to come to your site?
If you are just indexing headlines from other sites, then why should someone visit your site over Google News or the hundreds of other news aggregators?
Be different, stand out from the crowd, and then focus on marketing - your time will be better spent!
Quality content is the unique key to get authority backlinks and traffic especially for news related sites. They’ll copy your news article and put your link at the bottom of article as a original source.
I try to be different but less people know, till now…
Thanks for advice.