Lyme can cause an increase in fibrin, creating a hyper-coagulation of the blood (also known as thick blood), and if he was dehydrated, this will also complicate issues.
I've always been hard to get blood from. My veins get most nurses excited (LOL!) as they look all plump and easily accessible, but they are far from it! My veins roll and fold over themselves, meaning one needle stick, if not done carefully and correctly, will mean up to 4 or 6 holes in my vein.
The best thing to do is insist they use a butterfly needle (
/goo.gl/o1qd48) or a child's size needle, and to be sure that he is well hydrated. If this is still an issue, you can have his blood tested (Prothombin time) to see how fast his blood is clotting and you can apply heat (like a heating pad) to the arm where they will be drawing blood just prior to the blood draw (minutes before).