Would the Mongols have attacked Mecca? What would be factors that may cause the Mongols to attack Mecca?
First of all that might happen only in ATL, I presume.
The obvious POD is the later death of the Great Universal Khaan and so Hulagu doesn't have to withdraw most of his forces into Mongolia (thus making the Mamlyuk victory possible).
So in this ATL the Mongols conquered Syria and Egypt.
And the Arabs are in between Iran and Egypt - two main possessions of Hulagu.
To secure communication lines between those two regions Hulagu might think that the Arabs have to be taught a lesson, something spectacular, awe-inspiring. For example sacking Mecca. To show everybody who is in charge and that even the deserts cannot save you from the Mongol wrath.
But that's not too probable.
The mentality of the Mongols of that period is 'do not piss off (any) gods without good reason or by necessity'; and sacking the sacred town might piss off the Muslim god from their point of view, which should be avoided if possible.
(Ye, the Mongols sacked Baghdad and murdered the caliph, but they considered him to be more secular, than religious leader.)
And last but not least - what saved the Arabian towns deep in Arabia for millennia - too much effort, too little gain. So you don't sack Mecca for the reasons of profit, it's too protected by nature and it's not too rich.
I guess the Arabs, closely associated with Mecca have to infuriate the Mongols like killing the Mongol ambassadors or someone important, a Borjigin of high birth, that's possible as the Arabs were often engaged in raiding. And when it is a point of honor, the revenge - then it is more likely that the Mongols would make such a difficult campaign for so little profit.
Also, what would have been the effects of an attack? What would have been the reaction of the world?
Even in OTL the Mongol conquest was a shock to the Muslim world as it was. In the ATL it is even more. So sacking of Mecca wouldn't change anything too much.
As I said the Mongols would't hurt sacred Muslim sanctuaries in Mecca even if they sacked the town, they were too superstitious ))