Brief hospital course (BHC) summaries are common clinical documents generated by summarizing clinical notes. While large language models (LLMs) depict remarkable capabilities in automating real-world tasks, their capabilities for healthcare applications such as BHC synthesis have not been shown. To enable the adaptation of LLMs for BHC synthesis, we introduce a novel benchmark consisting of a pre-processed dataset extracted from MIMIC-IV notes, encapsulating clinical note, and brief hospital course (BHC) pairs. We assess the performance of two general-purpose LLMs and three healthcare-adapted LLMs to improve BHC synthesis from clinical notes. Using clinical notes as input for generating BHCs, we apply prompting-based (using in-context learning) and fine-tuning-based adaptation strategies to three open-source LLMs (Clinical-T5-Large, Llama2-13B, FLAN-UL2) and two proprietary LLMs (GPT-3.5, GPT-4). We quantitatively evaluate the performance of these LLMs across varying context-length inputs using conventional natural language similarity metrics. We further perform a qualitative study where five diverse clinicians blindly compare clinician-written BHCs and two LLM-generated BHCs for 30 samples across metrics of comprehensiveness, conciseness, factual correctness, and fluency. Overall, we present a new benchmark and pre-processed dataset for using LLMs in BHC synthesis from clinical notes. We observe high-quality summarization performance for both in-context proprietary and fine-tuned open-source LLMs using both quantitative metrics and a qualitative clinical reader study. We propose our work as a benchmark to motivate future works to adapt and assess the performance of LLMs in BHC synthesis.