I've looked through the Monster Manual for clues but I can't seem to find anything concrete about how much a pteranodon (or any other flying creature for that matter) can lift while flying. They have a fifteen- to twenty-foot wing span, and are medium beasts (MM, pp. 79–80) so it seems reasonable to me that they could carry a medium-sized PC. But that seems like it might be a bit overpowered. Is there any rule for how much a flying creature can lift in 5E?
There is no rule that applies any stricter carrying capacity for flying creatures, so you'd default to the standard carrying capacity on p.176 of the PHB.
Picking up and carrying a character would fall under push, drag or lift, so it would be 30 times the pteradon's Strength score, or 360 lbs. At that weight, the creature's speed is reduced to 5 ft. Below 180 lbs., it can move at its full speed.
According to the rules for flying creatures on p.191, a flying creature drops if its speed is reduced to 0 or if it's knocked prone. Since the pteradon in this case has a speed of at least 5 ft., it can still fly until it's carrying more than 360 lbs.
You can also apply the optional Encumbrance rules from p.176, which will reduce the pteradon's speed more incrementally, but the general rule remains that as long as it has Speed, it can fly.
There is no rule.
The "push, drag or lift" phrase assumes that the creature is standing on solid ground. There is nothing in the text to suggest that it was intended to apply to flying movement, and the physics of lifting a heavy object while standing is entirely different than the physics of flight. A locomotive engine, for example, can easily pull a train up a hill but would be utterly incapable of powering an airplane with a fraction of the weight.
The proof is in the calculation. The strength-based response suggests that a Pteranodon can lift roughly four times his weight. To compare - we know that eagles can carry roughly one third their weight. Unless we assume that Pteranodons, like dragons, fly with magic, the result is absurd. If you desire a quick-and-dirty house rule that gives a reasonable answer, I suggest simply taking the earth-bound answer and dividing by ten.
Use the carrying capacity rules on page 176 of the Players Handbook. According to that, it would be able to lift 360 at 5 ft movement, or 180 at full speed.
As for whether you can ride it, the answer is yes, but only if you are small. As per page 198 of the PHB ‘Mounted Combat’, a willing creature that is at least 1 size larger than you that has an appropriate anatomy can serve as a mount. Since pteranodon are only medium, you must be small (or smaller) to ride one. Also, the pteranodon must be willing to let you ride it. So a gnome Druid could summon one up (technically 8) with conjure animals, and then command it to let you ride it.
The idea that there is no special rule is supported in a (now unofficial) tweet by Jeremy Crawford:
Q: What are the rules for creatures carrying friendly PCs? e.g mage polymorphed into giant eagle carrying allies? (5e)
JC: See "Lifting and Carrying" in the Player's Handbook (p. 176) for rules on carrying capacity.
With the followup tweet:
Q: Are we to assume no adjustment for flying vs walking?
JC: The carrying capacity rules make no distinction between walking and flying creatures.