Your player is wrong. Empower only affects variable numbers which are represented by the die. The number of missiles created by the spell based on caster level is not variable.
The correct way to apply empower spell is to take the damage, in this case its (4(1d4+1))*1.5.
This is further clarified in its interaction with maximize spell
An empowered, maximized spell gains the separate benefits of each feat: the maximum result plus half the normally rolled result.
Benefit: All variable, numeric effects of an empowered spell are increased by half including bonuses to those dice rolls.
Saving throws and opposed rolls are not affected, nor are spells without random variables.
A missile of magical energy darts forth from your fingertip and strikes its target, dealing 1d4+1 points of force damage.
The missile strikes unerringly, even if the target is in melee combat, so long as it has less than total cover or total concealment. Specific parts of a creature can't be singled out. Objects are not damaged by the spell.
For every two caster levels beyond 1st, you gain an additional missile - two at 3rd level, three at 5th, four at 7th, and the maximum of five missiles at 9th level or higher. If you shoot multiple missiles, you can have them strike a single creature or several creatures. A single missile can strike only one creature. You must designate targets before you check for spell resistance or roll damage.