Answer:
water is Arthur's favorite drink
Step-by-step explanation:
The three drinks are lemonade, milk, and water. 
If Paige's drink is a dairy product, milk would be the only answer there.
If Frank's drink comes from a fruit, lemonade would be the answer. Lemonade is made from lemons and sugar. 
Water is the only choice left and Arthur is the only person who doesn't have a drink. 
So, Arthur's favorite drink would be water.