Thanks for reaching out
The cover letter is a crucial step in your application.
I would say there is no check list to be met in the cover letter, just some basics to include. Here are the three most important topics to include :
1 - why you want to work in the consulting firm you are applying to: you have to be very specific about the firm, otherwise the consultant/HR who will screen your CV and cover letter won't find it differenciating. Try to talk to 3 to 4 people in the firm you are applying to, from different seniorities in order to get a sense of how this firm is different from the others, and use their insights in the cover letter (do not hesitate to mention a few names)
2- why you are different from the other applicants: here, you need to describe your academics, your extracullicular activities, potentitial internships and business related activities. Be specific but to the point . Mention the experiences that make you stand out the most.
3- Describe what you are looking for in an internship experience in a consulting firm : why you are applying, what you want to get out of it and how it fits into the company's culture. Try to talk to a current/former intern from your school to get insights.
Again, the more unique you letter is, the more people will remember you.
Last but not least, be concise, statistics show that the screening process takes 3 minutes maximum. Better stand out.
Best of luck in your application and do not hesitate to reach out to me by private messages if you want specific advice.