A while ago I was thinking of just this question. It's nice to find a thread to give me an excuse to put my thoughts in writing.
I was particularly pondering whether U2 could lay reasonable claim ahead of the Stones. My thinking was that U2 have remained a watched for recording act for 30 years. Their albums still are a bit of an event when released. While I thought most except die hard fans stopped watching for Stones albums after maybe Tattoo You. So that their influence as a recording/writing band could be restricted to around twenty years, with the last thirty basically running off the fumes of that period.
Having pondered that though I still in the end would give the Stones second place. Primarily on the strength of having such a large range of classic rock tracks. It's hard to beat a back catalogue of Satisfaction, Jumping Jack Flash, Gimme Shelter,Paint it Black etc etc. they can fill a whole concert with songs that most people, even those not fans, would know. And being one of the two biggest acts of the 60s would put them higher on the influential rankings than a later era band.
And in a way the fact that they've been able to stay a huge touring act on the strength of the songs from that original 15 to 20 year period is a testimony to how substantial their catalogue and contribution is.
While U2 has some great songs and some that will be considered rock standards, I'm not sure it's enough to get them that second spot. As well, only releasing 12 albums in thirty-odd years also loses them a few points in my book. I know for large acts a three or four year gap is now the norm but its hardly the way to be seen as a prolific contributor to rocks repertoire.
U2 can fight it out for minor placings with the Who, Zep, Pink Floyd and the Kinks.