My mother always has the same problem. Always did. She can't
   donate. When she has to give her own blood to get tested, there
   is a smaller needle you can demand - and she's had to be
   demanding because some of them INSIST on using the fat needle.
   She also discovered she can get blood out of the back of her
   hands with the small needle. She says, "I have small veins. I
   need the smaller needle. Also, take it from my hand". It's
   amazing how they don't listen and "want to try" anyway... like
   she's a test subject. Anyway, they end up failing and get
   someone more experienced... after they poke a few holes in her.
   Feel bad for her. One answer that helps her is drinking a lot of
   water the night before or the morning of. Puffs up the veins.