My 86' Raleigh 10 speed with friction shifters insists on not holding the lowest gear. If I hold it in the low gear, it will stay but the moment I remove my hand, it goes back up. I'm assuming it is a bent derailleur because it theoretically shouldn't resist a gear that much... I have fiddled with all the adjustors, but nothing is changing the problem.
It may not be a bent derailleur, but a bent derailleur hanger. You can buy the park tool derailleur alignment gauge and bend it back to shape or have a shop do it for you.
After that I would check the cable tension, disconnect the rear derailleur cable and make sure that the chain stays on the smallest cog to start and then slowly the tension appropriately to make sure it shifts in to all gears properly.
Also, check to make sure you don't have a loose bolt on your friction shifter that is allowing too much play in the lever, which may let it slide out of the desired position.