# On the basis of faith in his name

"Because of faith in his name" or 2) this is how the man was made strong.

# On the basis of faith in his name

"Because we believed in Jesus's name" 

# his name

"his name" refers to either Jesus or for Jesus's power.

# made this man ... strong

"made this man ... well"

# The faith that is through Jesus

"The faith that comes from Jesus" or 2) this is faith in Jesus.