Sorry, but your question is confusing: Flash Player is an Adobe product which allows online video streaming. Your DSC-V1 is just a camera and neither has nor needs Flash Player.
Perhaps you're confusing it with the camera flash used to provide light when taking photos? If the flash is set to "on" it should usually pop up and fire except when the camera is set to certain scene modes (see below). The other flash settings are "off" (no lightning flash icon) and "slow" (the letters SL appear next to the lightning icon). The only other icon which can appear next to it to the left are the scene modes. These only appear when the mode selector dial is set to "SCN" and there are six of them: beach mode (palm tree and beach icon); snow mode (snowman icon); portrait mode (a pair of heads and shoulders icon); landscape mode (mountains in a picture frame icon); twilight portrait mode (crescent moon and a head and torso icon); and twilight mode (crescent moon icon). Of these, only the last one prevents the flash from firing even if switched on.
You can learn more about these by clicking this link
to the Sony website which has a downloadable .pdf manual. The scene modes and what they do is explained on pages 63 & 64.
I hope that you've found my answer to be of use and in return ask only that you return the favour by rating my answer. If your problem remains unresolved then please explain clearly what you'd like to know in a comment and I'll try again: I've been using a DSC-V1 regularly since 2004 so there's not much about it which I'm unfamiliar with.