This function should have a docstring saying what it does. That follows the form of:
"""One line summary

Multi-line description.
"""

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This is a bit complicated. I'd suggest breaking it up:
q = User.objects.filter(...)
q = q.distinct().order_by(...)

return q[:5]

Do you need set()? I would expect distinct() to take care of this.

The other pages (for review UIs and the diff viewer) will also need this.

_beginEditField is too generic a name for this action.

Missing a doc comment.

Blank line after this.
Also, you need some scoping so this doesn't search the entire page. You can use this.$('.suggest_user')

You should re-query here. Instead, in render(), store a reference to this element, and use that here.

Only one var statement per function, at the top, like this:
var allPeople = ...,
    i;

Blank line before this, and remove the trailing whitespace.

There's some alignment and style guideline issues.
For example, you should have spaces around operators (-, ==, etc.)
Also, in JavaScript, comparison should almost always be done with === and !== (the == and != versions do casting, and are slower... yes, it's weird).

A lot of the same comments about field references, variables, etc.

I don't think setEditFieldValue is a thing.

You can just use show() and hide() instead of setting a class for elements.

A lot of the same comments as above with variables, references, style guidelines.
This seems to me that it does the same thing as _relayoutSuggestions above. Am I right? If so, we should just be using that function.
Also, really, this function should not know about specific field types. Instead, we should probably have some new, optional methods in the entries for defaultFields that are called here when editing begins (beginEdit above) and when it ends (complete), so that the field itself can control this UI.

Blank line before this.
Single quotes.
===

This line doesn't need to be removed.

Some general comments:

The HTML tags should be indented relative to their parent tag.

Template tags have the {% part unindented, and then spaces between that and the template tag name indented 1 relative to the parent template tag, like:
{% if foo %}

{%  for x in y %}

{%  endfor %}

{% endif %}

Ideally, <br /> shouldn't be used. Instead, make the thing below in a <div>.

CSS class names should use - instead of _ as separators.

Can you go through the current review and mark issues as resolved if your new diff addresses them?

I think my update indents correctly now, but correct me if I'm wrong.

I think this is doing a lot of unnecessary work. The .distinct() call should make the set() unnecessary, and converting to set/list will evaluate the entire queryset when all you care about is 5 objects.
QuerySet implements slicing in a way that will avoid extraneous database access, so you can just do this:
return q[:5]

I was doing that before, and it didn't actually give me unique values, but the use of set was just a temporary hack so I could send something that appeared sensible to the front end... I plan to redo most of this method soon.

This looks unused?

Did you test what happens if you click on suggestions both while the edit box is open and when it's not?
I'd like to see specific test cases you executed in the testing done, instead of just "behaves as I expected"