The radius of the bubble is approx. 150ly.
That means the surface area of the bubble (4πr²) is about 282600ly².
If we divide that by the 10ly range then I think we have around 28260 systems that could be current starting points from which we could begin our colonisation efforts.
This is a complete guesstimate but let's say each one of those has 2 or 3 systems within 10ly that only it could colonise.
That means there's something like 70,000 systems up for grabs on day 1.
Someone PLEASE check my maths but that sounds like a lot of systems to dismiss as "uninteresting".
OK, checking the maths:
Only one noteworthy error I reckon - you can't divide an area by a distance and get a dimensionless number like a system-count. Instead, you need to divide the area by another area (e.g. the average area on that sphere per system).
I was going to guess somewhere between 5 and 10 ly for the distance between systems out there, but then realised that I could cheat and get a better estimate of it by using Spansh to count the systems between 150 and 152 ly from Sol (it gave 952). This is probably including pretty much all of the systems in that sphere, I guess, since there are probably few or no undiscovered systems in the vicinity of the Bubble.
That yields an average inter-system distance of roughly 8.4 ly, using what is I guess a bit of a fudge - I just took the cube root of: (the volume of that 2-ly-thick shell divided by the system count within it) - but is probably plenty accurate enough for this purpose.
Using that to estimate a per-system area would give us around 4000 systems on/near the surface of the sphere, most of which probably do have some neighbours which could be colonised.
We can of course cheat
even harder and just use Spansh to directly give us a estimate (again this depends on the assumption that everything remotely near the Bubble has been discovered already):
First of all, adopting Ian's note that the Bubble radius is more like 200 ly, and then just asking Spansh to give us all systems between 200 and 210 ly from Sol, it returns 7978 systems. All of those are
just outside the perimeter of that 200 ly sphere, and are thus likely to be (just about) colonisable from within that sphere. (The number is very linear with the thickness of the shell, so if we choose 8 ly we get almost 6400 systems etc.) This is a pretty good match to Ian's guess of "probably under 10,000 for actual outward expansion". (Who's surprised?

)
If we just use the 8.4 ly system separation on its own, and use that to very roughly estimate the expected number of systems in a 10-ly-thick annulus around a 200-ly-radius sphere, i.e. ~ 4*pi*200^2*10/(8.4^3), we get an estimate of 8480 systems, which is the same as 7978 if you're a physicist
(In fact, the average "known to Spansh"-system separation, calculated as above, is a wee bit higher at 200 ly from Sol than at 150 ly - around 8.
7 ly - which
may be a hint that Spansh does
not know about all systems in that annulus... Or it may just be a random fluctuation in the star density.)
So who wants to guess at many Elite players are going to want to colonise a system each Thursday morning?
I have no real instinct for this (others will!), but it's pretty hard for me to believe it'll be even close to 8000, especially after the initial novelty-surge wears off. If I had to guess, I'd say after the first few weeks it'll be < 1000 but I look forward to finding out.
(The recent
very attractive CG got almost 23000 participants, but I'd say colonisation is a lot more niche than "I want the awesome FSDs".)