I think this is already being done; it's called Social Media.
The reality is that maintaining social relationships take a lot more than just intelligence; in the past, these relationships were also limited by the fact that people needed physical proximity in order to relate to each other. The phone expanded our ability to maintain friendships and the like over long distances, but it's tools like Facebook that seems to be truly successful at doing that. I think that's for 2 reasons.
Firstly, photos and videos. That visualisation means people feel like they are truly part of the action. Secondly, social media limits the experience and puts it into an 'interrupt driven' mode; what that means is that you don't get the sense of touch or smell through facebook but we don't seem to really need that to be 'social', or at least one form of it. Also, the other person can post when they have the time, and you read when you have time. Even if these times are disjointed, it doesn't feel like that to the person reading the post.
If we take the limited form of socialisation that sites like Facebook allows as being a new form of social interaction, then for Millennials at least, their Dunbar Number has already risen. For those of us from older generations, we're less likely to lose contact with someone who moves away and therefore our Dunbar Number is also increasing through better retention of existing relationships.
To address comments...
Yes, there are some studies that appear to show that the Dunbar Number is unchanged by sites like Facebook, but (in my opinion) they fail to recognise a fundamental fact about the way the human mind has been changing since the invention of the Gutenberg printing press; we are re-wiring ourselves to focus less on memory and more on processing and analytical capability.
This is even more so since the dawn of the internet. Why remember heaps of facts when you can just look them up via Google or Wikipedia? This is especially so in a world where those 'facts' are changing so fast as our understanding of science and other subjects is growing so quickly, rendering old facts obsolete faster than we can learn them.
We no longer ask our children in schools to name the capital cities of a hundred nations. We ask them questions like 'why are many capital cities NOT the economic centre of their nation?'
We're testing their ability to analyse and 'fact-check', not remember.
When we consider this trend, I'd argue that if you take the classical model of the Dunbar Number, Facebook et al are actually decreasing our Dunbar numbers because people no longer find it necessary to 'remember' everything about their friends; they have a handy reference available to them at all times. Further, I'd argue that this is the reason most Millennials prefer to talk to friends on Facebook rather than organise large social gatherings; they have access to the information they need to interact with each other in the medium in which they choose to interact.
Whether we like it or not, we're using our computers and smart phones as artificial memory. We go to them whenever we need to know something, rather than remember it directly. That frees us up to do the real value added work of analysing the information we retrieve from these devices.
In that sense, (with respect to @JBH's Answer which is very good) we ARE increasing our width and depth of memory as a species through artificial means. We're effectively reserving our biological memory as a 'cache' which we fill from our artificial memory, retrieving the relevant information at will, as needed.
In that sense, research that involves surveys to determine Dunbar Numbers misses (in my opinion of course) the one inalienable fact about the way we process information today, including social information; the average person out there CAN'T remember enough to increase their natural Dunbar Number, but that's not what counts anymore. In the world of social media, people now have selective Dunbar Numbers, meaning that they can selectively remember what they need to about a much larger cohort of individuals on an 'as needed' basis, thanks to what amounts to artificial memory.