The emergence of large language models (LLMs) provides an opportunity for AI to operate as a co-ideation partner during the creative processes. However, designers currently lack a comprehensive methodology for engaging in co-ideation with LLMs, and there is a limited framework that describes the process of co-ideation between a designer and ChatGPT. This research thus aimed to explore how LLMs can act as codesigners and influence creative ideation processes of industrial designers and whether the ideation performance of a designer could be improved by employing the proposed framework for co-ideation with custom GPT. A survey was first conducted to detect how LLMs influenced the creative ideation processes of industrial designers and to understand the problems that designers face when using ChatGPT to ideate. Then, a framework which based on mapping content to guide the co-ideation between humans and custom GPT (named as Co-Ideator) was promoted. Finally, a design case study followed by a survey and an interview was conducted to evaluate the ideation performance of the custom GPT and framework compared with traditional ideation methods. Also, the effect of custom GPT on co-ideation was compared with a non-artificial intelligence (AI)-used condition. The findings indicated that if users employed co-ideation with custom GPT, the novelty and quality of ideation outperformed by using traditional ideation.