Just about anything can work - I don't really think one thing is better than other here. The thing to focus on is using them, not the actual lubricating. Exercise the seacocks once a month or so and you won't need to lubricate anything. So for now, get them loose - add some leverage if needed - spray some silicone lubricant or even something like PB Blaster to help them moving. After that, just make sure to give them some regular exercise.
I believe non-petroleum based was recommended as petroleum could deteriorate the soft goods in the valve. As Dennis stated, frequent exercising of the valves is important.
Found this on the Forespar Marelon site
QUICK NOTE ON MAINTENANCE New valves come greased. Valves that have been sitting on store shelves for a long time may need more and have the handle/ball exercised. Use any non-petroleum grease (Lithium, Silicones, etc) and no aerosols! Use a soda straw to get the grease through elbows and tailpipes and USE THE VALVES regularly to avoid marine growth from building up and seizing them. The more you use them the less you have to maintain them!
If you have the original Yellow handled Conbraco Apollo valves. They require no lubrication. Just as mentioned above. cycle them every so often. I have been the Conbraco Marine Rep for 31 years. Grady switched to Marine Hardware valves about 6 years ago, due to a buying group decision.
my 18 year old Grady has the Conbraco valves and they still work perfectly.