If BigCo could create a pseudo-incubator where the evaluation process was similar to YC. The company could filter out applicants whose ideas didn't align with their business objectives.
Additionally, BigCo could own the rights to the product (or some portion) while the "founders" get salary/benefits for their time. Perhaps reduced salary but with large performance benefits. Really, the monetary issue wouldn't interest me as much as the ability to work on a project where I had autonomy and interest. (I wouldn't apply with project ideas that I had no interest in)
Actually a friend of mine interviewed for a "startup" that's in the same position. The company has little fear of going under from bankruptcy, but they have the startup feel with the random basketball hoop, video games, snacks, etc that bigco doesn't offer to their employees. They have autonomy to do their own iteration and that's what's preferred by the parent company.