You can use STUFF to insert a string into another string. It deletes a specified length of characters in the first string at the start position and then inserts the second string into the first string at the start position.
For example, the code below, replaces the 5 with 666666:
DECLARE @Variable NVARCHAR(MAX) = '12345678910'
SELECT STUFF(@Variable, 5, 1, '666666')
Note, that the second argument is not a string, it is a position and you are able to calculate it position using CHARINDEX for example.
Here is your case:
DECLARE @Variable NVARCHAR(MAX) = '[some useless info] Useful part of string'
SELECT STUFF(
@Variable
,CHARINDEX('[', @Variable)
,LEN(SUBSTRING(@Variable, CHARINDEX('[', @Variable), CHARINDEX(']', @Variable) - LEN(SUBSTRING(@Variable, 0, CHARINDEX('[', @Variable)))))
,''
)