My employer currently offers a traditional 401k (tax deferred contributions and gains) but being rather young, I'd prefer to have a Roth 401k (post-tax dollar contributions and tax free withdrawals upon retirement). It doesn't cost a company anymore to offer this benefit (not considering employer matching contributions) but it could be somewhat of an administrative burden, especially to initially set it up.
How do I convince my company that this is something they should do?
Is your company small enough for you to speak with the decision maker? If not, your best bet is to contact the company that offers your plan. If you can contact your company's point of contact or broker of record, that would be best.
I must be missing something here.
Why don't you just open a Roth-IRA? Since you are young, you are probably under the $90,000 income limit to open one. Tax wise, it will have the same effect as a Roth 401k.