1 day in Beijing, what do you think?

Definitely Yonghegong Lamasery (aka Lama Temple). I would pick as well Wangfujing night market and hutong tour. Or, if your tour is bringing you to only Badaling Great Wall, you might want to consider visiting another section as well.
The Kunming Lake in your tour itinerary is Summer Palace, I'm sure.

Only go to the Lama temple if you are interested in temples. A lot of people do like it but frankly it did nothing for me at all, just another temple with a huge buddha. I would say the hutongs if you aren't doing that on your other tour. Also you didn't mention any of the markets for shopping. If that isn't included on your organized tour (which would surprise me) then somewhere like Hongqiao market (beside the temple of heaven). The HouHai lake area is nice to wander around at night. Wanfujing night markets are interesting although the food there is over priced in the extreme. I always take visitors there to see it and take photos and then go somewhere else to eat.

If the free day is your first in China you could also just enjoy wandering around (the area to the west of Houhai is good for this) and getting acclimatised.

I would sugest also a visit to Jingshan Park (north of the Forbidden City) - if you are there on the weekend, it is interesting to see quite a bit of interesting activities - singing troupes, dancing classes, martial arts practice....etc. You can check out my blog with info and photos from that at Sites in Beijing, China

I'd say Lama Temple and hutongs, like #1 said.

I really enjoyed the summer palace.

#5, Summer Palace is already included in the tour itinerary under a different name -- Kunming Lake.
Cheers!

Since you will be seeing lots of beautiful lakes, try something somewhat different and see the Hutongs and Lama Temple.

Unfortunate the Confucius Temple and Imperial College halls are closed for renovation.

Good luck.