Almost there..
members.filter(**{'string__contains': 'search_string'})
To understand what it’s doing, google around : )
Understanding kwargs in Python
**
expands dictionary key/value pairs to keyword argument – value pairs.
To adapt your example to the solution:
variable_column = 'name'
search_type="contains"
filter = variable_column + '__' + search_type
info=members.filter(**{ filter: search_string })