But no idea where the 'y' comes from.
Is is comparatively recent..
From the OED:

"ploidy . Biol.
[f. ha)ploidy, poly)ploidy, etc.]
The number of homologous sets of chromosomes in a cell, or in each cell of an organism.
1947 Genetics XXXII. 512 A state of indefinite ‘ploidy’.
1953 Jrnl. Gen. Microbiol. VIII. 101 Another possibility is that there is a different degree of effective ploidy of the F+ and F- gametic cells, the F- gametic cell having a higher degree of ploidy (or, possibly, more nuclei) than the F+.
1961 Lancet 5 Aug. 318/1 All metaphases have been analysed. When an exact account was impossible the ploidy level has been estimated.
1970 Watsonia VIII. 140 As most of our counts are approximate the ploidy level rather than the chromosome number is given.
1976 Nature 29 Apr. 785/2 To confirm the ploidy, both strains were grown axenically and their chromosomes were stained."

my impression is that when only haploid and diploid cells were known, there was probably no need for a category term.