Just a few ideas for how to improve the eidolons power and customizability a little bit without making any major system changes. I wanted to see what people think of them. I'm going for concept, so the exact implementation of these might need some work.
Idea 1: Weapon Based Attacks
Instead of having the primary attack be 1d8 with a feat to add a trait to it, make it based on a martial one handed weapon.
Choose a single one handed martial weapon. Your eidolons primary attack gains the damage die and traits of that weapon. You choose the weapons damage type between bludgeoning piercing and slashing. If the weapon is versatile you choose the second damage type as well. Thrown and any other trait that is not compatible with the anatomy of a natural attack do not apply.
This would allow the eidolon's attack to be customizable without needing to create a system for building it. Power wise this would only add a single trait to d8 attacks, which seems reasonable. Additionally, a feat could be added to upgrade the attack to a one handed advanced weapon, or 2 handed martial weapon.
Idea 2: Action Evolutions
One of my favorite things about PF2 is how many of the martial feats give the characters special actions. The eidolon's evolution feats don't have enough of these. While the evolutions availabe are good to have as options, I was hoping for more action based evolutions. However, just making eidolon copies of the standard martial feats would be boring. Instead, I think they should be based on monster actions. For instance based on the Hydra and the Maralith:
Many Limbed Assault [A][A][A]
Your eidolon lashes out with every available limb at once. The eidolon makes a single strike with its primary attack. This attack deals an additional damage equal to 1d4 plus and additional 1d4 per 2 levels you possess.
Its nice to see that the dragon gets its breath weapon and draconic frenzy, but not every creature that people will want to make an eidolon based on will be available as a base form, so adding abilities possessed by multiple monsters across multiple types allows the player to approximate these creatures without making the system completely free form.
Idea 3: Feat Granting
One of the problems that the eidolon runs into is that while it shares the summoner's skills it does not share skill feats. I understand the balance concern of being able to share feats like battle medicine (effectively doubling how often you can use it on a target), so I think as an alternative summoner's should be able to choose, when they take a skill feat, whether it will apply to them or the eidolon. This would come with an extra rule that whichever body gets the feat is the one that has to qualify for it, allowing the eidolon to take intimidating prowess without the summonner needing high strength.
Additionally this feat granting could be extended to general feats and class feats used to take an archetype. This would have to come with restrictions on what the eidolon can gain from these feats, such as no weapon or armor proficiency, no spellcasting, and nothing that the eidolon would be incapable of using, such as the magic items granted by scroll trickster. I can't think of anything that would make for a better dread marshal than a giant monster that is till chewing on the last enemy it killed.
Idea 4: Gestalt Synthesis
I think the current synthesis feat could use less restrictions than it currently places on the summoner, but I like the idea of having a feat that allows the summoner to choose whether or not they want to merge with their eidolon. However for those who want to be in eidolon form all the time I have a solution that I think works better than wearing the eidolon as a suit. This would be a class path alongside Master Summoner and Eidolon Caller or whatever they end up being called if paizo decides to add class paths in addition to the eidolon selection.
Gestalt Synthesist
You haven't just summoned a creature from another plane of existance, you have become it. You have completely merged with your eidolon, turning into a singular creature, and nothing can separate you any more. Instead of gaining an eidolon, you fain all of the abilities that your eidolon would have possessed, including unarmed and unarmored proficienies. You use your own ability scores, not the eidolons. You can select Strength or Dexterity as you primary ability score in place of Charisma. If you take any evolution feats, they apply to you.
This allows for all the fun of playing a monster built on PC rules, without the ability score shenanigans of the original synthesist. This basically leaves you with something like a magus, but with eidolon abilities instead of Striking Spell and, ironically, Synthesis.