To the end of having more users provide a name or a nickname, I think one will have relatively more success if one communicates that a (for some and/or some practical purposes) also a displayname that is not the real name but something more easy to recognize and memorize than the names provided by default has some value.
So, if one wants users to choose an actual displaynames, first, one might encourage to do this giving some motivation for doing so in the faqs, and not only insist on the real names issue (in particular, giving a reason that is in my opinion is a complete anti-reasons; even if I wanted to use my real name [I don't], I never would given what the faqs say on why one should, but let me not digress).
As discussed, the sign-up process could be more geared towards gving one (but there are gobal considerations against this on meta.SO see my comment for a link)
One thing that is actually done in this direction is the autobiographer badge that encourages giving the pieces of information asked for in the profile.
Contacting by email was already mentioned, but personally I think (if this is feasible technically, but I guess it cannot be that hard in pronciple) a message/notification/reminder on the site would be more efficient. Say, when making ones third post without yet having changed the displayname one gets a notfication/reminder on the site regardings this with some infor why it could be good. (The advantage is that then people just might do it right away.)
But it should also be recognized that some actively want to avoid being recognized via displayname. (This applied half-way to me when I started using the site, though it was also somehow an accident I did then not have any actual displayname, which I changed when it became inconvenient for practical reasons).
In brief, if the goal is more people with actual displaynames one should not only insist on realnames (one still can, even at the same level of intensity, though it seems even a bit unnecessary as meanwhile it is hard not to note this is the dominant culture on the site) but also the actual subject raised in this question should be recognized somewhere (and not, as in various contributions here already, be right away conflated with the realnames question).