Currently my summoner's aquatic eidolon is wearing an amulet of mighty fists and a belt of strength. What other magic item slots besides neck and waist does an aquatic eidolon have available?
Because eidolon evolutions can result in a creature with bizarre anatomy, the RAW has no definitive rules about its magic item slots. The GM has the final decision.
The rules for a summoner's eidolon list several base forms that are the basis for the creature's appearance and anatomy. Every base form with several free evolutions. The avian, biped, quadruped, and tauric forms have limbs. That means that aquatic and serpentine forms initially have no limbs (unless you separately add them).
According to the table of Magic Item Slots for Animals, creatures such as fish, manta ray, and walrus are categorized as having the "piscine" body type. The category seem to fit the description of the eidolon's "aquatic" default shape. These creatures only have a belt slot, an eyes slot, and a chest slot for a saddle. You could easily argue that this seems too restrictive for an eidolon.
However, luckily for you, written above the table is the following text:
Additionally, GMs may use this table as a guide to determine what kinds of magical gear non-humanoid monsters can wear and use. Note that the rules in this section are merely suggestions, and ultimately it is up to the GM to decide what kinds of animals can use particular types of magic items.
The game's designers did not expect eidolons to fit a category of animal shapes, and so there are no RAW listings of the creature's magic item slots. Therefore, the GM decides which slots are available.
Based on personal experience, I recommend that the GM determines slots based on the eidolon's shape evolutions. Example: if you describe your eidolon as having a neck and head, then it could have the neck and head and/or headband slots. If it gains the "Limbs" evolution, then perhaps it gains wrist slots.
One thing to keep in mind: According to the eidolon's Link ability, if the eidolon wears a slotted magic item, then the summoner cannot personally wear an item in the corresponding slot. This limitation also means that the eidolon cannot gain additional item slots (for example, an eidolon with two heads does not benefit from two headbands).