Find this translation in a TM. Translation memory is always the first place where you can find the translation you need. Everything that you have accumulated on similar projects could be useful. We keep some TMs from LSPs we worked with. For direct client we maintain the TMs on our side.
If you found a suitable translation in a TM, ask a native linguist to confirm that it's the best one possible. You have already done a part of work for them, and they can simply reply with a quick email. We do not ask for linguists' help only in one case—if the current translation context fully coincides with a previous similar project. In any case, the PM never comes up with something on a whim and translates anything themselves.
Add this free translation request to another order. Check the list of current projects and try to find the one which would match your request by customer, subject and language pair.
If the project is still at the translation stage, simply send an additional file to the linguist and add the right amount of words to the PO. The project budget will not suffer much from this. You would pay much more if you sent a separate request to the linguist and issued a minimum fee PO for it.
In case the files have already been bundled up into relevant folders and the option above won't work, you can also come up with a suitable solution. For example, I ask linguists to translate the necessary phrase at the LSO stage, which is paid by the hour. Most likely, the linguist will have enough paid time for both the main task and the small additional request. If it is not enough—we will pay extra time without any questions.
Ask the linguist directly to do a translation for free. This can work only if the linguist is someone you work with on a regular basis.
Write directly and be honest. I can ask for such help only from those linguists with whom we have long and trusting relationships. In this case, I write everything as it is: "Dear John, we received a small request for translation—just three words. Unfortunately, we do not have any budget for this project. I would be grateful if you could help me at no cost."
Do not forget that the linguist also has the right to say "no". I always end such requests with the phrase: "Please inform me if you believe this request should be paid." When you show that you respect the linguist's time and give them an opportunity to decline, there is more chance that they would agree to help you.
Do everything you can to ensure the request doesn't take much time. If you write "my request will not take you more than a couple of minutes," it should be so. Prepare the request email so that the translator does not have to spend extra time opening programs and reference files. Write directly in the email: what needs to be translated, consistently with what other prior translations, and what term should be used. Write briefly and clearly so that the linguist can open the email, read the instructions and reply immediately without additional clicks.
Do not overuse the linguists loyalty. They are not obligated to do anything for free—they are just doing you a favor. For you, asking them to do something for free is an exception, not a usual way to save money.
Once a linguist lectured me for a free translation request: "When there are many free work requests in a row, it does not feel fair." As it turned out, a couple of days before my request he had already helped my colleague at no cost. This is a good example of how the wish to save money can badly affect your relationship with your linguists.
Remember the risks. Free work always carries risks. Any linguist, no matter how good they are, may not be as diligent when they work for free, as they are with paid requests. If this happens, you won't have a leg to stand on blaming them, but the client will be equally dissatisfied with errors, regardless of whether the order was paid or not.