Is there a name for the abbreviated (usually three syllable) title? Surely the act of abbreviating an anime title to three syllables has a name...? And what is that name? (some of these will show my age)

╔═════════════════════════════════════════════╦═══════════╗
║                  Full Name                  ║ Shortened ║
╠═════════════════════════════════════════════╬═══════════╣
║ Kodomo no Omocha                            ║ Kodocha   ║
║ Jungle wa Itsumo Hare nochi Guu             ║ Hareguu   ║
║ Ore no Imouto ga Konnani Kawaii Wake ga Nai ║ Oreimo    ║
║ Fruits Basket                               ║ Furuba    ║
║ Japanese Animation                          ║ Anime     ║
╚═════════════════════════════════════════════╩═══════════╝
For what it's worth, in Japanese, you would call this particular sort of abbreviation a 略称 ryakushou or 略語 ryakugo, both of which basically just mean "abbreviation". There isn't an anime-specific term for this sort of thing or anything. 

I should mention that abbreviated names of anime are more commonly four syllables than three-syllables, at least in recent years. "Hareguu" is Ha-re-gu-u (actually, these are morae rather than syllables, which is why this may not be obvious if you're unfamiliar with Japanese); "Oreimo" is O-re-i-mo; and Fruits Basket is frequently "Furubasu" Fu-ru-ba-su in Japanese. There is a nice table of these on Nicopedia (JP).

As a side point of interest, this blog (JP) collates abbreviated names for anime of recent seasons - for example, 2015 winter, spring, summer, and fall. (Obviously, machine translation is going to fail spectacularly for this sort of thing, so this is only relevant if you can read some Japanese.)