There wasn't any major invasion seriously planned or launched during the Crusades, but that's not particularly surprising for a couple of reasons, which I'll go into in a moment. The closest the Crusaders came was when Reynald of Chatillon launched a raid on Mecca in the mid-twelfth century (I can't remember the exact year). The raid was quite small, only a handful of ships, and it basically raided down the Eastern coast of the Red Sea and failed to actually reach Mecca. While this raid clearly had an impact on Muslim observers who feared of a possible greater attack, it was strategically a non-event and mostly just stirred up anti-Crusader sentiment with no real gain. Anne-Marie Edde discusses the raid in her biography of Saladin and places it in the broader context of his military career.
There were two main reasons why an attack on Mecca and/or Medina wasn't particularly feasible:
Neither city was particularly well located for a military campaign. If you look at a map, Medina is almost as far away from Jerusalem as Baghdad, and Mecca is even further away. The land between the Crusader States and Medina was also not particularly hospitable, so any campaign attacking the cities would almost certainly have had to sail down the Red Sea. With no crossing point for ships from the Mediterranean the ships would have to have been built in either the Gulf of Suez or the Gulf of Aqaba, neither of which were bastions of Crusader strength. Building a fleet in enemy territory with minimal supplies was a feat beyond the means of the Crusading armies, and even if they had succeeded the Red Sea was still Muslim controlled territory and venturing down the coast would have been quite hazardous.
It's not clear how great an understanding Crusaders had of Islam and its holy cities. While Reynald's raid strongly suggests he was at least familiar with the holiness of Mecca, how widespread that understanding was is not clear in the sources. Most of the chronicles we have from the Crusades are shockingly ignorant of Islam's beliefs, possibly a little too ignorant. It's very possible that Crusaders would have had a greater knowledge of Islam than our sources imply. However, they would also need to have received Western support (either financial or military, or both) to launch an invasion of the Arabian Peninsula and convincing Westerners of the importance of Mecca and Medina may have been harder.
The only thing really in favor of this kind of invasion would have been the fact that neither Medina nor Mecca were particularly significant cities politically or militarily at this time, especially when compared to Baghdad, Damascus, or Cairo, so if the Crusaders had managed to get an army to either city they would not have struggled to take them. However, actually moving those armies, and surviving the Muslim response to a sack of their holiest cities, would have been a likely insurmountable challenge. Suggested Reading:
Anne-Marie Edde's Saladin contains a lot of interesting information on Islam during the 12th century.
Amin Maalouf's The Crusades Through Arab Eyes is a bit dated, but also the most approachable book on Islam and it's relationship to the Crusades.
Thomas Asbridge's The Crusades is probably still my favorite general history of the subject.
There wasn't ever an effort to try and conquer the cities, though Raynald of Châtillon, the knight who famously caused the devastating defeat of Saladin's forces at the Battle of Montgisard sent several crusader 'pirate' ships into the Red Sea from the Kingdom of Jerusalem sea fortress of Helim (modern Aqaba, Jordan) with the intention of threatening Mecca and Medina. They never successfully reached the cities however and Saladin's conquest of Jerusalem and ceasing of Helim meant that the crusaders never had the opportunity again to reach Mecca by sea (and of course, land was never an option; the route would have been fairly perilous for the crusaders, and most importantly fruitless - the crusaders had their forces stretched thin enough as it was, without any natural form of reinforcement. The focus of the crusaders between the 1st and 3rd crusades was nearly consistently on protecting the held territories and, on several occasions, ultimately failed attempts to invade Egypt).
Edit: I should note, Raynald did sack several villages along the Red Sea coast, and came very close to Medina before being captured. He was eventually executed.