I would go by word of mouth and reputation.
I would want the tutor to analyse all aspects of where my child is at maths, not just add/sub mult/div but all topics and explain how a Sats grade has been given on their current knowledge.
I would probably want one-to-one. We did a one-to-four and the tutor didnt have time to deal with all four children with different needs and was still charging £25 per hour for each child.
I would want them to start at the very basics and be sure they understood every step of the strategies/concepts they are learning. Many children can do harder maths but not really understand how they got there.
I would want lots of practical, physical hands on learning, not worksheets or computers. They can do computer work at home.
I would want to see written evidence in workbooks about what they have learnt, so that they have something to refer back to.
I would want honest feedback - ability, behaviour, attitude.
I would want someone that can relate to my child and that my child is comfortable with, in a pleasant working environment.
I think an hour can be too long, so it would have to be broken up into lots of fun games and activities. If the child is happy with the environment then they can cope with an hour.
Good luck with your adventure, I hope you get sorted.