I lived in Lisbon for a few months, I loved it, and I think I could be of help here.
In my opinion, Avenidas Novas would fit your description “somewhat trendy, with good places to eat and near public transit” - this is a modern city area well-served by transport (unlike most historical ones) as it was built in the 20th century. I was able to work from a different wif-fi café/bakery almost every day, and never covered them all (check Linha d’Agua for example), an endless choice of local and international food and easy access to the party town (10-5 min via metro). My favorite spot was a park (Gulbenkian) with high speed wi-fi where I could work “in the nature” while the shopping center is 5 min walk. Another positive is that as this is the business City, you get things open until late.
Second best, I’d say São Bento - walking distance to party, the trendiest restaurants in town on your doorstep, again loads of wi-fi cafés. Downside is the transport - its an area built on a steep hill with the nearest metro (Rato) being on top of that same hill. However, compared to Avenidas Novas this area feels much more authentic with its tramways, 19/20 century buildings, markets. It is expensive but your indicated budget would be OK pretty much anywhere in Lisbon.
Alcantara, as somebody already mentioned, is great for nomads because of the LX factory and its under-the-huge-bridge appeal but for a longer stay you might get bored of LX and Alcantara is somewhat isolated from the city. Also, public transport is affected by mass tourism - you do have the trams but you might wait for 3-4 cars before you squeeze in. I prefer the relatively quiet metro station and trip from “Parque” million times.
If you feel adventurous (maybe for 1 week of your stay), have a look at Casilhas - its on the opposite side of the river Tagus (Tejo) and it is served by regular transport of a Portuguese kind - iconic orange boats which take 8 mins to get you to the Lisbon port, just in front of the bar area. I lived on the other side for a while too and travelling by boat twice per day (for 2.30 return!) with the great view this trip offers, tops all public transport in the world. It really makes your day. Living in Casilhas also means you have a selection of fish restaurants on your doorstep, and an awesome view of the city. It’s 2x+ cheaper. However, last boat is at 1.40 am and you might get stuck on the wrong side of the river later than that (taxi circa 25 euro). Nothings works late and the wi-fi cafés are limited edition.
My recommendation - Alcantara, Casilhas or Alfama for short (romantic/cultural) stay; Sao Bento/Rato or Avenidas Novas for longer term city life with easy access to anything.
This is obviously my view - if you need anything else, let me know.
P.s. I’ll be in town for the Web Summit in 2 weeks.