Specifically, does a Quadruped eidolon add its STR, or 1.5*STR on its bite attacks?
Assume for the sake of this question that the eidolon has at least one other natural attack and that it's not using any weapons to attack.
The PFSRD describes the bite evolution like this (emphasis mine):
An eidolon’s maw is full of razor-sharp teeth, giving it a bite
attack. This attack is a primary attack. The bite deals 1d6 points of
damage (1d8 if Large, 2d6 if Huge). If the eidolon already has a bite
attack, this evolution allows it to deal 1-1/2 times its Strength
modifier on damage rolls made with its bite.
Earlier on that same page, we see that a quadruped eidolon gets the bite evolution for free. However, it also lists a bite attack in the base statistics. It seems like there are two possible cases:
- The eidolon has a bite attack because it naturally has sharp teeth. Additionally, the eidolon gets the bite evolution for free, which means its bite attack adds 1-1/2 times its Strength modifier on damage rolls.
- The eidolon receives the bite evolution for free, granting it a natural bite attack. Because the bite is a primary attack (but not the only natural attack, meaning it doesn't add 1-1/2 anyway), the eidolon's bite attack adds its Strength modifier on damage rolls.
Rules-As-Written is most preferable, but it seems as though the rules aren't clear, so a justification of Rules-As-Intended or even What-Makes-The-Most-Sense is appreciated. If you don't have a justification for one of these two possibilities, feel free to recommend your own interpretation or sensible house rule.
None of these
Normally, an Eidolon only has the attacks provided by its evolutions, so the quadruped form has a bite attack in its stats block because it starts with the bite evolution for free, and has no other attacks. That bite, due being its only attack, will already add 1-1/2 times its Strength modifier on damage rolls, as noted in the same page.
If you later purchase other form of attack (claws evolution, as a example), or gains somehow another natural attack (as you asked us to assume), the bite attack would add only 1x Strength modifier from that point onwards, since it is not anymore its only form of attack. This is when the part you quoted from the bite evolution enters play. Purchasing the bite evolution again will allow your Eidolon to apply the increased modifier always, even if its bite attack is not its only form of attack. But the free bite evolution represents the attack with what the Eidolon was created, if it gains more attacks, it would be later.
As a less serious note, if you go by option 1, then you quadruped eidolon, having 4 legs due being naturally quadruped plus 4 more legs due the free limbs evolutions, would start with a total of 8 legs, and it would not be a Quadruped anymore.