One full day/night in Paris - Where to stay?

Consider staying in the very center of Paris. If you stay on Ile St. Louis, you'll be right in the heart of Paris and about as centrally located as can be. It's right behind Notre Dame, a metro stop is close by, and there are several good places to eat on the main street.

There are a number of hotels on the island. We've stayed at Hotel des Deux-Iles and were very pleased. They also have a sister hotel on the island.

If you decide to stay on the island, we can tell you about a several places to eat.

If it’s cool you want, I’d look in the haut Marais, around the Marche des Enfants Rouges. So many restaurants and bars, and few tourists. Or the area in the 11th around the Oberkampf metro.

If it were me, I would want to stay somewhat near the train station from which I would depart (assuming you are taking a train to Lille). That way you can stop off at your hotel right before departing. Although the area right near Gare du Nord hasn't been recommended by many on this forum, I stayed two nights at Hôtel Maison Mère last winter—which is a fairly short walk to the station (15-minute walk, I ended up departing from Gare de l'Est right next door). I remember it being a fine hotel at a reasonable price, and a nice, family-oriented neighborhood. I usually walk almost everywhere in Paris and I was able to walk to Notre Dame, Montmarte, several covered passages (great for shoping) and Rue Montorgeil (great dining) easily enough.

Set the Samaritaine department store at Pont Neuf as your goal on the map and search lodging very close to it.

https://www.dfs.com/en/samaritaine

for instance:

https://www.hotel-ducs-de-bourgogne-paris.fr

What kind of "late night" are you looking for?

Bars, late night food, bands, coffee…just something to do when it might be others bedtime.

The 10th or 11th arrondissements.

Since you're only staying one night, the nightlife depends a lot on the day of the week. Thursday through Saturday are the best days (after 11 pm, after dinner). Maximum opening hours for bars and restaurants are set by law. With a few exceptions and specific permits, restaurants stop serving after 11 pm, and bars must close by 2 am at the latest.

The 11th arrondissement, in the area around Rue Oberkampf, is a very lively area with a vibrant local scene of bars and restaurants in the late night.

The OP’s message says that is a Friday night so a good one for nightlife. However, as it’s your first international trip you may be underestimating the likelihood that jet lag will limit your plans for late night action. It’s as much as most people can do just to stay awake until dinner time their arrival day in Europe, so I would prioritize what you think you may do during the day when planning where to stay rather than ambitious night plans. I am a bit curious why you’re leaving Paris so quickly for Lille. Not that there’s anything wrong with it (or so I hear, I’ve never been), but it’s not a city at the top of very many folks’ European bucket list.