This is a pretty difficult question to answer: not because of the availability of information, but because of the parameters of the question. What do you mean by rare? What do you mean by amiibo? Are we including any and every amiibo made by Nintendo or a partner, or are we only talking about amiibo that were commercially available to the general public for a time?
If you’re looking for ways to obtain these amiibo for significantly less than their usual prices, consult Doing amiibo on a dime.
Let’s assume these three parameters: we’re only talking about official amiibo made by Nintendo and its partners, we’re excluding any one-time or very limited edition amiibo that weren’t directly available for sale, and we’ll define rare as the single amiibo least produced and sold to our knowledge.
Given those three parameters, the most obvious options are the Qbby amiibo, the Mega Yarn Yoshi amiibo, and arguably the Mii Brawler amiibo.
Qbby was only produced in limited quantities and only sold in Japan. These days it goes for around $300, and seems to steadily climb each year. While the price could spike with another addition to the BoxBoy! series, we could see Qbby fall from this list if Nintendo ever reprints the amiibo to replenish its supply.
Mega Yarn Yoshi
Initially sold at $40, Nintendo was obviously not expecting Mega Yarn Yoshi to be an in-demand product. It’s a very large version of the regular Green Yarn Yoshi with the same functionality, so Nintendo didn’t see fit to produce many of them. Given the inordinate amount of cuteness as well as the lack of reprints, Mega Yarn Yoshi is usually worth around $200 at the time of this writing.
Obey the Yoshi.
Originally sold in the Mii 3-pack (which I managed to obtain at launch), Mii Brawler has, for whatever reason, become significantly more valuable than its two Mii counterparts. Mii Gunner is around $30-40 out of box, and Mii Swordfighter is about $40. Yet Mii Brawler, if anyone is selling one, tends to clock in about $80. There’s no sales figures that say there’s a small supply of Brawlers, but the price indicates that there’s a significant disconnect between the demand for Brawler and the supply of him, so we can still conclude there’s fewer Brawlers on the market than the other Mii Fighters.