Under Section 51 of the New York City charter a commission draws New York city council districts after the decennial census. The redistricting rules in the charter are ranked by importance and include restrictions on the number of people, geographic boundary length and width, and a fun puzzle to minimize the length of the sum of the boundaries. The next round of new boundaries will be in place in time for the February 2023 City Council elections.

Department of City Planning Bytes of the Big Apple

Under the section "Political and Administrative Districts"

https://www.nyc.gov/site/planning/data-maps/open-data/districts-download-metadata.page

NYC Open Data

https://data.cityofnewyork.us/City-Government/City-Council-Districts/yusd-j4xi

Doesn't include council member names, just district numbers.

New York City Council

https://council.nyc.gov/districts/

Doesn't provide spatial data but does include a friendly interactive map with council member names and links to individual member web sites.