How did the activities of protocells become inheritable?
The sequence is believed to be opposite of that implied by the phrasing of the question: the inheritance of structures supporting inheritance came before protocells.  The important point is that you need a system of reproduction before building more complicated structures (such as a cell).  Without replication (reproduction), a cell would be little more than a soap bubble, and would disappear without a trace before long.

The initial (or very early) form of replication is thought to be based on a simple form of RNA. The reason for this is that (1) RNA does encode genetic information in a way that is replicatable, and (2) RNA can take on shapes and chemical structures that act as an enzyme (called a ribozyme) that catalyze chemical reactions.  The hypothesis is that an RNA molecule formed that catalyzed assembly of copies of RNA chains, including itself.

After such replication was established, probably some kind of enclosure like the cell membrane evolved to contain the chemicals, making the reactions more efficient.  After (or maybe before) this, RNA catalyzing assembly of simple protein-like molecules whose sequence was encoded by RNA sequences evolved, leading to protein-based cellular machinery.  This mechanism is still evident in life today in that protein assembly uses a ribozyme-based molecule assembly (the ribosome) to translate messenger-RNA into proteins.