No, you don't have to pay if you wish to change the name of the candidate in the GATE application form. Indian Institute of Technology, Bombay has extended the date for filling the application till October 7, 2020 without any late fee. So the correction window will be opened after the registration process gets over. Most probably, by the third week of October, 2020 the correction facility may be started. However it will be notified through the official website. So keep checking the official site for furthur updates.
http://engineering.careers360.com/articles/gate-application-form-correction
Different Dates will be allotted for modifying various parameters such as Name, DOB, College name, location, Roll no., Parents details etc can be modified when the correction window will be opened after the registration process gets over. As of now, November 13, 2020 is the last date for modification of details such as Category, Gender, PwD status, Exam paper, Exam city etc. Check out the above link for detailed information.