Getting into the Red Sea with an army is tough enough - Pharaoh Necho's old canal was, I understand, blocked even by the end of the Roman era. You would need to portage ships across from Port Said to Suez and get them and a well-armed force down the Red Sea to Jeddah (the port for Mecca). Assume a relatively-unopposed landing, your force would then have to cross about 70 miles of desert and semi-desert before besieging or taking Mecca. Hostile terrain in physical terma as well as in the face of pilgrims ready to die to reach the virgin houris in Paradise.
Lots of assumptions piling up here already!
Ok, so you have breached the walls, fought through to the Great Mosque and reached the Ka'aba. That will probably mean literally hewing a way through hundreds of human beings. The priests might try to remove the Black Stone, but you foresaw that and now have seized it.
The stone may be a meteorite, but whatever it is, what will you do to it? Burning or smashing it may not be much good (if it's an iron or stony-iron meteorite, a tough nut to crack). So you take it and any wealth you can lay your hands on, get back to Jeddah and get back on your ships, probably with fewer men (assuming that the pilgrims haven't managed to storm the ships before you return).
What to do with the Sacred Stone? You've managed to thoroughly anger every Muslim from Iberia to Indonesia and from Zanzibar north to Syria. Take it to Rome? No, the Holy Father would regard it as pollution. Take it to Jerusalem? Forget that, Temple Mount is holy enough as it is. Dump it anywhere on land? Again, too much risk of recovery. You dump it either in the Red Sea or halfway between Alexandria and Crete, out of reach of the hordes of Islam. The plunder has to be melted down (the arabic inscriptions make it too clear what it is, otherwise) and split up. So must your army, returning to where they came from in secret, lest the priests of the Ka'aba find them all and kill them.
An unlikely enterprise, therefore, and of little profit. As indicated, it might lead to a Jihad against Rome.
What is it with this fascination with the black stone? Generally, it's quite unlikely such an operation would aim at deliberately carting off the other side's relics (assuming the people who'd do such a thing even cared enpough to find out the things existed in the first place - for all some Latins knew, they'd expect to find some dog-headed demon living there).
A much likelier scenario is: Crusader raiding party reaches Red Sea port in force, captures some ships. Leader orders men on ships, starts raiding other coastal settlements. Resistance is weak because he got inside the enemy's loop. There aren't supposed to be any Christian pirates in the Red Sea. Raiding party resupplies, draws more men, heads off towards Jiddah. jiddah is poorly defended and the attackers get in, standard massacre ensues. Mecca is now within reach, so off they go. It's standard operating procedure for every armoured thug from the Altavilla to Rainald de Chatillon, and it worked scarily often IOTL.