I was thinking about my homemade 3x8wt5 fixture. What value of cap should I use if I make this HPF and should the cap be wired before the power switch or after it? (Before means it would have constant powerto it.  After means it is switched on and off with the lantern.)0

I just did some mathing with the rapid tables calculator: http://www.rapidtables.com/calc/electric/power-factor-calculator.htm
Here is the input:
Real power: 24 watts
Current in amps: .538a (2x .19a and 1x .158a ballast.)
Voltage:120
Frequency:60hz.
Output:
PF: .372
Apparent power: .064560kVA
Reactive power:  .059933kVAR
Correction capacitor: 11.040129uF
Please tell me if my mathing is correct and if I need to fix any errors. Thanks.

Sorta corect....

 - Why are the ballast with so different currents ? Is there accordingly visible difference in the brightness of the lamps ?

 - The real power draw includes the losses of the ballasts, so probably on the order of 30W and not 24W

 - Not that it matters, but why the output power calculation in KVA ? Designing PFC for a lighthose that draws power in the KW range are we ?

One note A good oiled filled cap will take up space
In a a such a small lantarn for PFC. 
 
Second like ash said your not including Ballast loss
 
Third u might land on a odd valule for Cap sizes.

My experience with the self healing film capacitors is, most of the newer ones (90s & later) go down pretty bad with capacity as they age (power on hours count). A 10 year old capcitor may well be at 1/2 its rating....

A higher voltage rated capacitor should last way longer. For 120V a 240V capacitor would be good (you can use a 50 HZ rated one, it does not matter when you give it half the voltage)

Dont know what is the case with oil filled ones, i never seen one used as PFC in a lantern here

You dont have to be spot on perfect with the power factor.... Anything in the 0.9's is excellent. Even the capacitors are +-10% value. If you use a capacitor at its rated voltage, i think better in this application to err on the high side (to go a bit into capacitive power factor), so that it have longer way to go before it degrade to the low power factor condition on the inductive side

Sorta corect....

 - Why are the ballast with so different currents ? Is there accordingly visible difference in the brightness of the lamps ?

 - The real power draw includes the losses of the ballasts, so probably on the order of 30W and not 24W

 - Not that it matters, but why the output power calculation in KVA ? Designing PFC for a lighthose that draws power in the KW range are we ?

well the middle ballast came from a newer lantern and the advance ones are older, and I also noticed the middle lamp was visibly dimmer than the others so I investigated.  The site wouldn't let me change the units so that's why it is in kva. I guess my house could be considered a lighthouse. I just did the freezer test and the middle one, the ballast with the lowest current, had the starter Fire five times versus once or twice with the .19a advance ballasts.

I mathed again with ballast loss included and here is what I got:
Input:
Real power: 30w
Amps: .538
Volts:120
Freq.: 60hz
Output:
Power factor: .465
Apparent power: .064560kVA
Reactive power:  .057166kVAR
Correction cap: 10. 530451uF.

I was thinking of using this one since its small and it can share a hole with one of the sockets.

I cant say i like those. They look way too tiny for their rating. How about a standard cylinder shaped one, like in HID lanterns or bigger motors ? If the chokes fit behind the lamps that would fit behind the starter socket area

It looks like some motor capacitor (where the current is limited by the motor winding, so no power available to cause anby severe fire or so), but does not look like rated for direct mains connection.

The capacitor has to be UL listed for the direct mains connection, then even with just "125VAC60Hz" it would be good enough.
Compare to the one on your picture, the UL listed would be safer, more robust and I would guess even bigger...

The thing is, I'm not sure there will be many components available for such use: With low power, the power factor is usually not any issue, so the fixtures are left uncompensated, with higher power (e.g. CMH's) the ballasts usually contain a multi tap transformer, where the PFC capacitor is connected to some higher voltage tap (to better utilize the winding. so have lower losses).

ANd in any case the capacitor should be connected behind the switch: If it is permanently on mains, it means ~60VA reactive load (so zero power factor) on mains when the lantern is OFF, quite a nonsense, when the low power factor is, what you want to address...