I'm by no means a professional historian, but my understanding of it is that Egypt only appears "relatively stable" if you look on the surface. From its founding to its end, Egypt had gone through 31 different pharaonic dynasties, each one lasting from only years and decades to centuries. With this in mind Egypt hardly seems remarkable or stable compared to other areas.
Egypt too, went through many battles and eras of conquest, specifically before c. 3000 BCE when Menes united the two lands of upper (southern) and lower (northern) Egypt.
An important issue to remember is that Egypt was not a massive ancient civilization that existed for thousands of years, but one that slowly struggled and grew overtime. One important reason attributed to Egypt's endurance is that it did not have a lot of options when it came to building lasting structures. Sandstone was the rock of choice as it was bountiful however the only other option for poorer dwellings was a mud/reed combination which while effective as a form of cooling and able to be quickly built, remained a temporary structure (certainly not one that could span hundreds of years). This meant that where wooden structures (in western civilization) would have sufficed as an early means of creating noble/royal buildings being able to convey status and be easily fixed, mud was not. As such stone buildings began being created and added too from early on as it was the only real option to work with.
The durability of stone meant many things for many areas of life, especially religion. As is the case with many civilizations religion is a cornerstone, creating structure for the masses, rules and many other cultural customs. This in turn enabled Egypt to obtain an entrenched cultural heritage that was not easily erased, which could happen by neighbouring countries invading and razing.
There are many other reasons as to why Egyptian civilization lasted millennia, such as neighbouring countries and the impact they had through trade, war, religion etc. E.g the Hittites (Syria).