Sophomore Summer Unknown Boutique vs Name Brand Non-IB

Unless the no name B gives you very tangible, easy to explain experience, I would opt for the name brand (although there is no absolute). Reasons big banks or any other F500 likes to see big names include:

1. It's a vetting process. You proved you were successful in a competitive big brand recruiting cycle which means another large firm got a sense of your intelligence, acumen, soft skills, fit, etc. Kind of like they vouched for you.
2. Large companies all have a certain level of office politics. Knowing you have experienced that is a plus.
3. Training and structure - the large brands have developed training programs and tracks for you. So the next gig knows you went through formal training and were part of a staedy structure
4. Teams -the large brands will put you on a team. You'll have responsibilities and be accountable to the team. Might even have to present a project to the team lead.

All of these are good indicators of the type of employee they want.

The boutique may actually give you more responsibility (which is good). For it to be leveraged, you'll have to gain skill at communicating the value you brought to the role and company. That becomes hit or miss.