A point that I would like to make is that HN is non profit and so is HNI. So the analogy you give may not apply here.
You should respect the law and change the name or take the site offline until you receive appropriate permissions.
Please note that even licenses as GPL or MPL (that govern the Firefox code for example) give you permission to use the code, not the trademarks of the project. Ubuntu had to make a Linux package called "abrowser" ( https://launchpad.net/ubuntu/lucid/+package/abrowser ) simply because Mozilla didn't agree with branding their modifications as "Firefox".
It also occurred to me that hackernews.in is clearly a website aimed at Indians, so I don't know to what extent US law applies.
Anyway, it seems that the name is going to be changed, so this whole trademark point becomes moot. http://hackernews.in/item?id=375
Make no mistakes, I agree that its a great service. I would use it myself if the outstanding issues are resolved. But as I have said before, helping yourself to someone else's brand and work isn't very ethical regardless of its legal status. My views about that wont change even if pg agrees out of politeness.
The situation is not exactly the same, because you did start with a code snapshot that PG made available to the public. However, it's generally considered bad form -- as well as confusing to visitors and search engines -- if you use a name that is extremely close to the original site's name.
Things would be different if you created something in a totally different medium, or if, as others have pointed out here, you first got explicit permission from PG and YC to do this. The site name "Hacker News", though it is composed of two generic English words, does now form a unique reference to a specific site that carries a certain cultural cachet and set of expectations. To "ride" that, without permission, is not a good idea, even if you can get away with it.
One good thing that may come of this, however, is the notion that perhaps there's enough demand to make PG consider adding a "sub-Reddit" like feature to the HN codebase. So folks like you who basically just want to not reinvent the feature set, and also want to address an approximate subset of the core HN readership -- where in this case the subset is Indians or India-ophiles, etc. -- could do so freely with explicit support and functionality that enables it.
Copyright, IP and trademark don't only apply when profit is involved.
I reserve judgement on the appropriateness of appropriating the Hacker News name---but please don't labor under the impression that you can use someone elses (c)/(r) as long as you aren't making money.
Aside from seeking permission---you really should add a footer stating you are not affiliated with Y Combinator etc etc.
However, I thought I might point out the irony of all of this. A "decent" definition of Hacker is "a programmer who breaks into computer systems in order to steal or change or destroy information as a form of cyber-terrorism" (http://wordnetweb.princeton.edu/perl/webwn?s=hacker).
The irony comes when hackers get mad at other hackers hacking their hack.
I think this is called a cluster hack.