When you tap
Add new stopwatchon the main tab, Watches will create a temporary person (e.g. “Person07”). Then just tap on the watch to edit it, and tap on the name at the top, and type the real name. Watches will suggest existing names that match the one you've typed so far; if you see the one you want, tap it. If it's a new person, just type their name in, and Watches will create that person
You can also do this if you realize that you've entered the same person twice with slightly different names. Just edit the name of one to match the other. If there's data under both names, Watches will ask if you want to merge the two people into one record.