Hey guys, I received my Fairphone yesterday and was very happy, having been waiting for it for months (ok, with a sad eye also, considering the fact that I have to abandon my FP1 now for reasons of Mediatek’s failure to cooperate with Fairphone to render Android for the FP1 up to date)!
Today it really annoyed me when I was out and about, recognizing that I couldn’t manage to establish a working mobile data connection with it despite the fact that it had been switched on and my provider o2 had been found.
After coming home again I searched the internet, both here at the Fairphone Forum and the o2 forum. I found out that supposedly I also had problems with an APN that had not been configured correctly or configured at all. The information I found was not sufficient as it only mentioned what to fill in as “name” and “APN” and still left me without a solution for all the other parametres.
To make a long story short: I looked up the missing parametres in my good old FP1, copied it to my FP 2 and - lo and behold, it worked! 
Now I would like to share with you what worked for me, maybe someone else has the same problem with the same provider and might find my information useful. If not, then I can share at least my feeling of positivity with you. 
I hope I translated everything correctly from german to english.
Here are the parametres that helped me:
Name: o2
APN: internet
Proxy: not defined
Port: not defined
User name: not defined
Password: not defined
Server: not defined
MMSC: http://10.81.0.7:8002
MMS proxy: 82.113.100.5
MMS port: 8080
MCC: 262
MNC: 07
Type of authetification: not defined
Type of APN: default,supl,mms
APN ptotocol: IPv4
APN roaming protocol: IPv4
Carrier: not applicable
Type of MVNO: none
Value of MVNO: not defined
APN PPP phone number: not defined