Tel Aviv Friday and Saturday evening
If you're up for walking to the Shuk - then I'd recommend HaMitbachon (18 Rabi Akiva st.) over Haminzar for eating. Haminzar is a great bar where you can get some nice food - but it wouldn't be my first recommendation for a straight restaurant. I really like their fried sardines and sausages, but their "confit octopus" is one to miss. HaMitbachon is just off the shuk/in the Yemenite Quarter and a great mix of Ashkenazi and Mizrachi dishes and on Friday/Saturday they have specials like cholent.
Walking towards Jaffa along the sea you have Manta Ray which is a seafood restaurant that I've heard mixed reviews from CHers, friends (and have had myself, good and ok experiences) - but there's a restaurant further down the beach (closer to Jaffa) who's name I always forget but it's wonderful for seafood, mezze/salads, wine, etc. It's right off the boardwalk on the street side and not the beach side, very close to Jaffa.
A little further in Jaffa, you can just keep following the boardwalk into the port and there are a number of fish restaurants/bars (that I haven't tried but the atmosphere is really fun). I do really like Abdu Hadayag in Jaffa for seafood (37 Yefet Street), which is a bit further into Jaffa.
Another idea would be the old train station just to the south of Neve Tzedek (very close to your hotel). There's a pair of Spanish tapas style place (with the very kitchy names of Vicky and Christina), which also has excellent wine. Also in Neve Tzedek, there's a restaurant Dallal (10 Shabazi Street) that I believe (if I'm thinking of the right spot) is part of the Susanne Dallal performance space and is a truly gorgeous spot.
What is walking distance for you? You can walk to Jaffa and try Haj Khalil, or you can walk to the corner of Allenby and Gedera and try Haminzar, which is open 24/7. This is simple food, but well prepared. Lunch may be the better option, because it can get rowdy in the evening.
Another place you could try, one that gets good reviews, but I can't say since I haven't eaten there, is called Hakovshim, on the corner of Hakovshim St and Nehemia St. It is a bit closer to your hotel. I believe they are open on Saturdays.