The root word or prefix "syn/o-" means together, joint, alike, in conjunction. We can see this in the word synonym: "syno" meaning alike, "nym" meaning word. So its definition is a word that is like another, or are together nearly interchangeable. Medically, an example can also be seen in the condition syndactyly, a birth defect which two or more digits (fingers or toes, possibly both) are fused together; "syn-" meaning together, "dactyl-" meaning finger (or toe), "-y" meaning condition or state of.
The root word or suffix "-drome" means course or running. We can see this in the word palindrome. "palin-" meaning back (as in location), again, or repeated, "-drome" meaning running. So its definition is a word that when spelled (or "ran") backwards repeats the same original word.
Placing the two together, the medical diagnostic term syndrome actually means "running together" as in the co-occurrence of various symptoms during the course of a single diagnosis of a condition.