Other people's search strategies can be a useful source of search terms, and synonyms. I wrote in an earlier post that you should not just reuse a strategy. Using an existing strategy as a source of ideas is ok, but you need to understand how it works, and make sure that there is nothing missing.
But, of course, you might be updating a review. In which case, you would want to rerun the existing search. I think these are the issues to look for, assuming that you are not changing the research question or topic of the review.
1. Is there anything that should have been done differently? Is there anything missing? Is it ok to amend the strategy, just because you now see a better way to do it? At the moment, I think this is a matter for your supervisor, tutor, or someone with more knowledge of systematic review methodology than me!
2. Are there any new subject headings to include? MeSH is updated annually, and perhaps since the search was first run, there is a new heading. I think in these cases, it is ok to amend. For example, there is a MeSH term Patient Handoff/. It was first used in 2013. If the original search predates that, it will not include the term, so you ought to add it. What should you do to ensure you still find older references? How are they indexed? Checking MeSH (for example, in PubMed's MeSH database) may tell you.
3. Was the original search run in the same interface? Perhaps it was run in Medline via Ebsco, and you have access to Medline via Ovid. If this is the case, you need to translate the strategy and make sure that proximity operators and truncation are correct for the new interface. I wrote an earlier post about this sort of thing. It is also possible that the capabilities of the interface have changed since the search was first run, and this is a "translation" as well.