Doran's Ring is a cheap option which gives hp ap and mana regen. It is a good choice when your champ need only a little mana regen as well as some early ap, the extra hp also helps.
Chalice gives you massive regen, works well on champ with constant poke. Even better with the cdr from Athena. The magic resist also increase a bit of survivability.
Tear and RoA i will list under the same category. They gives a large mana pool. If your champ can burn through the whole mana pool in a fight (Anivia and Swain being the most obvious example), these seems to be the choice. Tear gives more mana and potential ap when fully farmed as well as the mana regen. RoA gives more survivability as well as sustain in lanes thanks to the passive. I personally favorite RoA over tear due to the lack of combat stat on tear and the cost efficiency of Archangel in early stage.