In addition to water and light, warm clothing, I would suggest bringing a light pack along in which you can keep a map and compass, insect repellant, sunscreen, a snack or two depending on the length of your hike, and a small first aid kit with the essentials (band-aids, antibiotic cream).
I agree that a small first aid kit is really important, including an ace bandage, just in case someone twists an ankle on a rock. I would also suggest a flashlight. You just never know. I've been on hikes that took longer than I expected, and I much appreciated have a small flashlight to get back to the car.
As for the mountain lion, making noise is the best way to keep lions at bay. many people in mountain lion territory tie bells to their outfits. I used to jingle my keys as I hiked when I lived in the Rocky Mountains. In general, it's also better not to hike alone, if it can be avoided. If you must go it alone, avoid early morning and evening times, as that is feeding time for mountain lions and they are most likely to be out hunting.
The mountain lion question is an interesting one because, in years past, lions have been spotted during summer time in our local neighborhoods as well -- including the gas station on Campus Drive.
My understanding is, it is important to resist any instinct you might have to run any. A lion would win a race with any of us, and running would just make a person seem like game to be hunted. Instead, make noise; try to seem as big and as unafraid as you can manage. Should the cat attack, be prepared to fight back, possibly using things like sticks or rocks to let you maintain some distance while doing so.
For mountain lions, I have heard that the best thing to do is stand as tall as possible, widen your arms & look as large as you can. It throws them off b/c they are used to their prey being on all fours. If you are with someone else, have then get on your shoulders to look even bigger. Put children on your shoulders, but be careful bending down so as to not look like an animal on all fours. And if they attack, attack back with large rocks, sticks, anything you can find around you. Try not to hike alone.