I see one article implying it’s a reference to the character from the movie Pitch Perfect.
Obviously, the fat part, which you understand, is because it’s a heavy airplane with a wide profile due to it’s intake ducts and internal weapons bay (rather than sticking the weapons externally where they will reflect radar like crazy).
Probably the same reason the P-47 Thunderbolt was nicknamed “The Jug” It looked like a milk jug from the side because the radial engine was so big.
The nickname for the A-10 Thunderbolt II (Same manufacturer of the P-47) is Warthog.
Combat aircraft have often been given female names, take the “Enola Gay” for example. It makes it easier for aircrew to deal with the reality of their job. In a similar vein pilots refer to the thermonuclear warheads they carry as “buckets of sunshine”.
Worth noting, as I understand it, “Fat Amy” is only referring to the F-35 variant that has VTOL capability. The other versions of the F-35 are not referred to as “Fat Amy.” I could be wrong though.