I try not to repeat myself! They all mean have a go in the sense of at least try to do something. Another one for the collection: Have a bash. They’re all said in a devil may care spirit suggestive of: Let your hair down - What the devil? Who cares? You’ve nothing to lose.
If you want something a little more formal, what about: Why not hazard a guess?
PS Your sentence seems to be up to par. Perhaps I would say:Could you please provide me with a few synonymous expressions for the [color=red]example given?