does the bionic arms bonus to strength increase your damage as well?

I would say as long as they're using that arm to wield the melee weapon, then yes. If they're using a two-handed weapon, perhaps not.

On two handers I would probably take the average bonus to strength across both arms. That is to say, its only a +5 bonus if its 1, +10 if its both.

I wouldn't let it be a +20 if its on both arms, as that gets a bit out of hand.

Actually, no. It is a bonus to Strength Tests, not to the Strength Characteristic.

Nathiel said:

Agreed, the wording is specific. +10 bonus when making strength tests, not to strength characteristic. It does help when trying to tear open doors or wrestle down an ork warboss though (since those are both tests).