A party member is considering riding my summoner's eidolon. Is my summoner's eidolon still able to attack whilst bearing a rider?
In short yes if it has the proper form and the mount evolution, but the fact that you're not the rider makes it complicated.
From the Ride Skill description
Guide with Knees: You can guide your mount with your knees so you can use both hands in combat. Make your Ride check at the start of your turn. If you fail, you can use only one hand this round because you need to use the other to control your mount. This does not take an action.
Fight with a Combat-Trained Mount: If you direct your war-trained mount to attack in battle, you can still make your own attack or attacks normally. This usage is a free action.
From Combat While Mounted
If your mount moves more than 5 feet, you can only make a single melee attack. Essentially, you have to wait until the mount gets to your enemy before attacking, so you can’t make a full attack.
This means that you can make a full ranged attack though with penalties at different moves. (double, quadruple, running. not quoted as it doesn't pertain to the question)
From the Eidolon ability (emphasis mine)
Eidolons are treated as summoned creatures, except that they are not sent back to their home plane until reduced to a number of negative hit points equal to or greater than their Constitution score
From the Summon Monster page
It appears where you designate and acts immediately, on your turn. It attacks your opponents to the best of its ability. If you can communicate with the creature, you can direct it not to attack, to attack particular enemies, or to perform other actions
All of these things in conjunction tell us a Eidolon gets all of its actions on your turn as you direct them It is still limited to basic action economy though (e.g: can't move and full attack, unless you give it pounce)
I don't know if there's any official ruling but I believe the Eidolon can only act on your turn, so your friend will be at your whim when it comes to his actual placement.
After searching for a bit i discovered an answer that was given to a similar question on Paizo's site:
Yes. A war trained mount is a weapon onto itself and standing in front of a sword wielding knight and a half ton beast with sharp hooves and teeth is a bad idea. If you have a sword this is easy, you and your mount full attack. If you have a lance and you're in range, you make your full attack, your mount 5 foot steps towards your foe, and then he full attacks. If you are right next to your opponent and have a lance, the mount makes a full attack, 5 foot steps away, and then you make a full attack.