How to filter an array of objects based on values in an inner array with jq?

Very close! In your select expression, you have to use a pipe (|) before contains.

This filter produces the expected output.

. - map(select(.Names[] | contains ("data"))) | .[] .Id

The jq Cookbook has an example of the syntax.

Filter objects based on the contents of a key

E.g., I only want objects whose genre key contains "house".

$ json='[{"genre":"deep house"}, {"genre": "progressive house"}, {"genre": "dubstep"}]'
$ echo "$json" | jq -c '.[] | select(.genre | contains("house"))'
{"genre":"deep house"}
{"genre":"progressive house"}

Colin D asks how to preserve the JSON structure of the array, so that the final output is a single JSON array rather than a stream of JSON objects.

The simplest way is to wrap the whole expression in an array constructor:

$ echo "$json" | jq -c '[ .[] | select( .genre | contains("house")) ]'
[{"genre":"deep house"},{"genre":"progressive house"}]

You can also use the map function:

$ echo "$json" | jq -c 'map(select(.genre | contains("house")))'
[{"genre":"deep house"},{"genre":"progressive house"}]

map unpacks the input array, applies the filter to every element, and creates a new array. In other words, map(f) is equivalent to [.[]|f].

Here is another solution which uses any/2

map(select(any(.Names[]; contains("data"))|not)|.Id)[]

with the sample data and the -r option it produces:

cb94e7a42732b598ad18a8f27454a886c1aa8bbba6167646d8f064cd86191e2b
a4b7e6f5752d8dcb906a5901f7ab82e403b9dff4eaaeebea767a04bac4aada19