There's no consistent way to trigger the Contingency before the damage
The fundamental problem, as I'm sure you've recognized, is that Contingency says (emphasis added):
The contingent spell takes effect immediately after the circumstance is met for the first time, whether or not you want it to, and then contingency ends.
Some generous DMs may allow you to get away with describing "creative" circumstances like "when I'm targeted with an attack/spell" in order to get the contingent spell to take effect before the attack's effects, but in general you probably shouldn't expect this to be possible. If you make the spell contingent upon an attack or other damaging effect, the spell will take effect after the damage is applied, which doesn't really help you, since you stated that you don't want to be knocked unconscious even momentarily, and because the temporary HP of False Life are not very useful if you're already at 0 HP. Furthermore, for your specific goal of preventing dropping to zero HP, there's no way to know if a particular source of damage will do so until the damage is rolled, which is too late: anything that triggers at that point will happen after the damage. Fundamentally, there is no timing that is both after the damage is known and before it knocks you unconscious. (This is the same reason that casting Shield requires logical, if not literal, time travel in order to work.)
Probably the closest you can come to something like this is making the spell contingent on a "hand signal" as you describe in the question and than manually activating it on your turn when you think you are likely to take enough damage in the next round to reduce you to zero HP. Obviously this won't cover every case, such as when you are paralyzed, stunned, etc., but no Contingency can cover every possible danger that might be inflicted on you. And even if you miss your chance to activate the spell before getting knocked unconscious, chances are you still have the opportunity to activate it after being revived, so it won't go completely to waste.
Another potential option, inspired by the suggestion from curious_penguin, is to use a trigger of "when I take damage equal to or greater than half my current hit points". The logic here is that if an attack deals more than half your current hit points, then a second identical attack will drop you to zero hit points, so this serves as a decent approximation of "one hit away from being knocked out". Of course, your DM might not allow a circumstance phrased mathematically in terms of hit points, so you may need to think a bit to come up with an acceptable wording that does the same job. For example, your DM might accept "when I take damage that I couldn't withstand a second time without falling unconscious" as a valid trigger, since it's a fairly common trope for characters to say things like "one more hit like that, and he's a goner". One side benefit of this circumstance is that it works for any source of damage, not just attacks. On the other hand, if you're up against a group of foes with widely-varying damage output, you run the risk of taking a series of small hits that don't deal enough damage to trigger the spell followed by one big hit that drops you straight to zero hit points before it can activate.
Ask your DM if "half HP or below" is a valid trigger
In D&D 4e, there is a concept of being "bloodied", which simply means you are at or below half your maximum health. For example, if your maximum hit points are 50, then you become bloodied any time you drop to 25 or fewer hit points. While in general your DM may not allow "below X HP" as a triggering circumstance, they might be convinced to allow a spell to trigger when you become "bloodied", since triggering effects upon becoming bloodied was a common mechanic in 4e, and if I recall correctly, "bloodied" was even specifically described as an observable state, the threshold between just a little bit scratched up to visibly injured (hence the name). In fact, the concept of being "bloodied" at half hit points, and of "bloodied" being an observable state, is also described in the 5e DMG's section on tracking monster hit points in the chapter Running the Game, though unlike in 4e it isn't given any mechanical significance:
[...] if a monster is below half its hit point maximum, it’s fair to say that it has visible wounds and appears beaten down. You can describe a monster taken to half its hit points as bloodied [...]
If your DM lets you use "when I become bloodied" as a circumstance, then you don't have to worry about manually triggering the spell on your turn. However, it should be noted that this can fail to protect you from damage that takes you from over half HP to zero HP all at once.
Consider triggering it upon receiving healing
Depending on how your party healers handle healing spells, it might useful for you to specify "when I receive healing" as the triggering circumstance. This works best if your healers tend to hold on to their healing spells until their allies are at low enough HP to be in danger of being knocked unconscious in the next round, since that's pretty much the exact trigger you're looking for. In this case, your contingent False Life acts as a force multiplier for the existing healing. And if the healers can't get to you, you can still manually trigger the spell by drinking a healing potion. The combined healing of the potion plus the spell might be enough to be worth using your action.
