Well... It's not quite possible to do so directly but we've come up with a workaround.

Why can't you just change context name?

• IssueSYNC still identifies remote contexts by their name, so changing it is a bit more complex. It was a early design decision, this is why it is s hard to move identify them by ID.

Will it be ever possible?

• Yes! We are constantly improving our product and making it more and more reliable and logically consistent. We will make it possible in some version but we will not make a big deal about it, although we will highlight it in release notes.

Workaround:

Assumptions:

• You are on production server
• You have already some issues synchronized within context you would like to rename
• You want to preserve synchronization with all already synchronized / linked issues.

Just play with JQL restrictions within contexts.
Our proposition is to add e.g. "created <= '19.07.2019'" to JQL restriction for context you would like to rename (same for remote one). Then create new context with JQL e.g. "created > '19.07.2019'" and link it with also newly created such in remote Jira. 

Pros:

• already linked / synchronized issues will have the same configuration (behaviour) as it did before the change
• newly created issues will have a new context name (which may be displayed for end users in comments, synchronization history or remote issue panel)
• adding such a contexts is about 2-5min of your time

Cons:

• old issues will remain to synchronize with old context name
• you will have two contexts for the same connection & project & issue type

*If your case is different than above assumptions just create new context and link it with remote on (it can be be one or old one).