Historically が and の had a much closer relationship. This relationship can still be observed in some modern words like 我が国, "our country". In this word が is a genitive marker and basically has the same function as modern Japanese の.
In modern Japanese が and の can't be used interchangeably, but の can still replace が in modifying clauses. Because languages don't like redundancy, there's still a difference between が and の as the subject marker of a modifying clause.
[...] It is because が emphasizes the word before it that this subject marker is frequently softened in modifying clauses by replacing it with の, a modifying particle that throws your attentions ahead. [...] (From: Jay Rubin (1998) Making sense of Japanese, Kodansha International)
By marking the subject of the modifying clause with の, you are saying "the wallet Mr Shimizu picked up is here" and its being here is more important than its being picked up from Mr Shimizu.
の implies that the information in the main clause is more important than the information in the modifying clause.
が doesn't imply the opposite though, it's more neutral.
The closest thing you can do in English is using a rising intonation.
清水さんが拾った財布はここにあります The wallet that (rise intonation) Mr Shimizu! (lower) picked up is here. - The implication being the wallet that is here is the one Mr Shimizu picked up.
清水さんの拾った財布はここにあります The wallet that Mr Shimizu picked up is here.