I think all of them are very determined, but if i Had to choose who the most determined is, I would say Blue.

Blue was kidnapped at the age of five and only knew her name and the fact that she was from Pallet. This girl raised herself and Silver into respectable people (after they stopped being thieves of coarse, lol). She has done so many things that she couldn't have done without her determination. She has gotten over her phobia of birds and caught the three legendary birds. She learned the ultimate move for water in FR/LG than Red and Green did so she could save her parents. She made it to the semi-finals in the Indigo League with one hit kills!

I Think all of them are tough all in their own way ! But the toughest ones are Blue and Kris!
Blue for the reasons the Espeongirl said!
And Kris because at such a young age and even after breaking her hands she finds a way to get stronger and learns how to kick pokeballs and capture pokemon and becomes an expert in such a short period of time :D And when she looses her confidence and has to get slapped by her mother to get back to her sences , she goes to train up the mountains and you can just see the wild determination in her eyes ! (That part when she tied her arms with bandages so she could simulate the training she had done when her arms were broken!)