I like to use chainsaw bar oil. sometimes I mix it 50/50 with STP if it needs to be a little thicker. I would not use grease on an outboad especialy a 3.5 it's just a bit more drag than I care to put up with in an OB.
I use 50/50 synthetic motor oil and STP. All the liquid lubes mentioned here are good. Fill the tube before every heat and you will not have a problem. It is easy to forget lubing the shaft when you are out just testing. Give it a shot everytime you refuel the boat.
I use a combination of oil and grease on my 3.5. I soak my flex shaft in a heavy synthetic oil by Pedro's then I coat the shaft with Pedro's synthetic grease to trap the oil in the braid. The Pedro's grease is a little stiffer than cold syrup and doesn't harm the Teflon tube over the long haul. I save the STP/oil mix for when I store my engines and flex shafts.
The nice thing about using an oil sump is that you do not have to take the shaft out at the end of the day. Just flush it out just like a motor and put it away. I took the shaft out 1 time midseason and it was liek new so I do not plan to take it out ntil the season is over.