Following are the Olympic sports that the ordinary Filipino can attempt to excel in without the heavy investment (save perhaps time, a training partner, and a trainer). Other sports could have made this list, but these require team training or heavy equipment investments. Most of the following aren't team sports, allowing the athlete to focus on oneself.
Badminton - A lot of training facilities. Off-the-rack equipment.
Boxing - We already have a strong boxing tradition.
Swimming - Akiko Thompson, Eric Buhain, et al. gave us a chance.
Table Tennis - Equipment isn't too heavy an investment.
Track and Field - Specifically the track events.
Triathlon - Swimming, biking, running.
As of this writing, fellow ASEAN nations Vietnam, Thailand and Indonesia have already won medals-- in weightlifting. Not a very "entertaining" sport, but our local sports commission should take serious note of this.