I would like to translate "Strong by Choice" to Latin as a motto for a charity. We have tried per arbitrium fortis but want to know if this can be improved.
Fortis is a good word for "strong". A harder problem is to find a good word for "choice" (or the verb "choose" if you are willing to reword a little).
Arbitrium means a legal judgement, a decision by an arbiter, an opinion, a decision, and similar things. A better option would be optio, which means free choice and the liberty to choose. (There is also the verb optare.) Another option is voluntas, which is more in the direction of desire, free will, and choice. The most appropriate tone depends on what suits your purpose. I recommend that you compare these words in any online Latin dictionary.
The correct form of fortis depends on what it refers to. If it refers to a single person, it is fortis (singular masculine or feminine). If it refers to a group of people, use fortes (plural masculine or feminine). If instead of people it refers to an institution, use forte (singular neuter).
The most idiomatic way to say "strong by something" is to put the something in ablative. Using per is grammatical, but I find that a plain ablative gives more gravity.
Therefore my suggestion is:
This leaves you with six choices, depending on the exact circumstances and nuance. The ablatives of the nouns voluntas and optio stay the same, no matter which form of fortis you choose.