No, it's not possible to extend the stay.
Q: What if I enter on the Visa Waiver Program and then decide I want
to stay longer than the 90 days?
A: You cannot extend the time on the Visa Waiver Program. The 90 days
also includes any time spent in Canada, Mexico and adjacent Islands.
Therefore you cannot cross the border into these areas and then return
for another 90 days. You can however ask for re-entry on the Visa
Waiver Program if you have left the continent.
However, you don't necessarily need to go to Europe. Central America, eg. Costa Rica, is also outside the VWP area and will probably be cheaper than flying back to Europe.
Your friend could have applied for a tourist visa, but if he's already entered it's too late. They can still go to eg. Canada and try their luck applying there; if they get it, it will override the previous VWP limit.