I wouldn't suggest the Imperial Palace or the Edo Tokyo museum for anyone's first visit with just 24 hours. However, if Kamakura has drawn your attention over and above all the other attractions of Tokyo, why not ?

As for where to go in the evening.
I used to love the club scene in Tokyo, music is fantastic. It is expensive but comparable to London yet with much cleaner, cooler venues and the ladies' toilets remain inviting well into the wee hours. No matter where in the world I've ever been I've always found it easy to connect with people when out dancing, female and male. YMMV.
Or...
Rather than bars why not sample food at the same time and go to an izakaya. Since I appreciate that language might be intimidating you could perhaps go to this place
http://www.andysfish.com/Shin-Hinomoto
It's not a British izakaya, it is an izakaya where the owner just happens to be British. Once you visit, hopefully you will feel more confident about popping into other izakaya in the same area, this part of Tokyo would be a lovely place to spend the evening and it's very likely you'll come across people you can talk to.

+1

Anybody up to date on the current club scene? What's popular these days?

The correct Japanese name for 鶴岡八幡宮 is 'Tsurugaoka Hachiman-gū' though it is sometimes referred to (especially on the web) as 'Tsuruoka Hachiman-gū'.

The Great Buddha and the Hachimangu Shrine are both popular but it seems a shame to go all the way to Kamakura and not visit some of the temples. There are too many to choose from but for convenience I would recommend Engaku-ji (円覚寺) which is close to Kita-Kamakura station, one stop before Kamakura on the Yokosuka Line. There are other temples nearby as you will see on this Japan Guide map.

Another well-known temple, Hase-dera (長谷寺) is not far from the Great Buddha.

The Great Buddha is inside of Kotokuin Temple (高徳院). :p

Yes. ;) Though it's nice not to have to share the place with busloads of tourists. :p

Last time I went to 円覚寺 it was during the cherry blossom season but once you walked 5 minutes from the entrance the place was very quiet. I walked from Kita-Kamakura via Engaku-ji and Kenchō-ji (建長寺) to Tsurugaoka Hachiman-gū, then along the crowded Dankazura (段葛) footpath to Kamakura station followed by some souvenir shopping along Komachi-dori (小町通り). The whole thing could easily be done in a morning. I could have gone to the Great Buddha afterwards if I wanted to.

Every hard core Kamakura fan needs to pick up a copy of Kamakura: Fact and Fiction by Iso Mutsu. Woefully out of date, but still the best way to understand the temples without becoming fluent in Japanese.

My favorite temple is Zuisenji (瑞泉寺), but way too far out of the way for a half day trip.

http://www.kamakura-zuisenji.or.jp/en/

Early October might be a little early for chrysanthemum season in Tokyo - but Mother Nature is never exactly on schedule. See what the story is when you travel. Flowers in Japan in season are quite beautiful.

Since you're staying at the Westin - I would perhaps check out the Rouge Bar at the Robuchon Chateau in your neck of the woods. It looked quite intimate to me (although it was closed at lunch when we dined there). Proper for a middle-aged or so woman as well. If I were alone in Tokyo - like on a business trip - I wouldn't hesitate having a drink there (and perhaps some food as well).

I really enjoyed the Edo Museum - especially the part that dealt with contemporary history (20th century). It's close to the shrine at Asakusa (must see IMO).

Also relatively close to the fish market (you can take a boat trip from near the fish market to Asakusa - it's a nice ride IMO - and you can rest your feet).

Also close to Kappabashi Dori - the part of Tokyo where there are lots of kitchenware/knife/plastic food stores. Excellent area to explore if you're into those kinds of things.

The things in that general part of the "clock" would easily be more than a full day for me. Robyn

There's plenty in Tokyo that doesn't look like Generic International Business Land, but taking a day trip to Kamakura is a good way to escape the central city. By this point, Kamakura is part of the Tokyo-Yokohama-Chiba megalopolis, but it is a small enclave with a different feel.

You might want to read up on the Kamakura Period (1185-1334) in Japanese history, when Kamakura was the shogun's capital. That will help you understand what you are seeing.

Kamakura is really worth the trek, IMHO the only "must see" destination in the greater Tokyo area. ;)

I did visit Enoshima and Kamakura and spent probably an afternoon in the area, like between 10am returning Tokyo by 6pm. Depending on the season, I would probably visit different sights, and when I visited, the Hase-dera gardens were in bloom during springtime. There are indeed plenty to explore in Kamakura but if you are hoping to explore in half a day, it is still possible to do so.

There is a link to a model course by Odakyu Railway here: Enoshima-Kamakura Model Course

And if you wanted to know more about my experience, I wrote a blog post about my stop there.

I like hikes but I'm not a big fan of temples as I grew up around them (and have been to the Buddha in Hongkong), so Kamakura was not very fun for me. I did enjoy the scenery of the Pacific Ocean though.

I like hikes and have visited Kamakura several times without visiting temples. I love the Pacific views, the 1950s (Showa) vibe of the town and the change in pace to Central Tokyo - quite a few surfers live and visit here. Had one of my best Christmas days in Kamakura; a delicious paella for lunch (a paella I would happily recommend to other arroz savvy Valencians) and we allowed our stomachs to settle watching Kites (the bird kind) circling and soaring at the beach. Very easy to spend an enjoyable day or half day in Kamakura without visiting any of the temples.