Iudex dicit honestatem plus valere quam mendacium.

Because after dicit, Latin is using indirect statement.

In English, we say something like The judge says that honesty is worth more than lying.
In Latin, that that-clause is usually expressed with:

• an accusative noun as the subject of the reported statement
• an infinitive verb

So instead of nominative honestas valet, Latin switches to:

• honestatem = accusative subject of the indirect statement
• valere = infinitive

A direct version would be:

• Honestas plus valet quam mendacium.
  Honesty is worth more than lying.

After dicit, that becomes:

• Iudex dicit honestatem plus valere quam mendacium.

For the same reason: it is part of an accusative-and-infinitive construction, the normal Latin way to report what someone says, thinks, knows, hears, and so on.

So:

• dicit = says
• honestatem plus valere quam mendacium = the content of what is said

Literally, Latin structures it almost like:

• The judge says honesty to be worth more than lying.

That sounds unnatural in normal English, but it is standard Latin grammar.

The indirect statement is:

• honestatem plus valere quam mendacium

Inside that part:

• honestatem is the subject of the infinitive valere
• valere is the verb
• plus modifies valere
• quam mendacium completes the comparison

The main clause is only:

• Iudex dicit = The judge says

Everything after that is what the judge says.

Because Latin normally puts both sides of a comparison with quam in the same case.

Here, the thing being compared is honestatem, which is accusative because it is the subject of the infinitive in indirect statement. So the other item, mendacium, also appears in the accusative.

A useful way to think about it is:

• honestatem ... quam mendacium
• honesty ... than lying

Since honestatem is accusative, Latin matches it with accusative mendacium.

Also, mendacium is a neuter second-declension noun, and its nominative singular and accusative singular look the same, so the form would not visibly change anyway.

Valere basically means to be strong, to have power, or to be worth depending on context.

With plus, it means:

• to be worth more
• to count for more
• to have greater value

So plus valere is a natural Latin expression for relative value, not just physical strength.

Here plus is functioning adverbially, modifying valere.

So it means:

• more

In other words:

• plus valere = to be worth more

Although plus has a more complicated grammatical history, at this level the simplest and most useful understanding is that it works here like an adverb.

Quam introduces the second half of the comparison:

• plus ... quam ...
• more ... than ...

So:

• plus valere quam mendacium = to be worth more than lying

It is the normal Latin word for than after a comparative idea such as plus, maior, melior, and so on.

Yes, you can think of one as understood.

Latin often leaves out words that are obvious from the context. Here the full idea is something like:

• honestatem plus valere quam mendacium valere

That is, honesty to be worth more than lying is worth

Latin does not need to repeat valere, because the comparison is already clear.

Yes, Latin word order is flexible.

This sentence is arranged in a very normal and readable way:

• Iudex = subject first
• dicit = main verb next
• then the indirect statement

Within the indirect statement, Latin places:

• honestatem first, as the main idea being discussed
• plus valere together, which makes sense as a unit
• quam mendacium after that, to finish the comparison

But other orders are possible, for example:

• Iudex honestatem plus valere quam mendacium dicit.

That would still mean the same thing. The endings, not just the order, tell you how the sentence works.

It would be:

• Honestas plus valet quam mendacium.

That is the direct version.

The changes are:

• honestas instead of honestatem
  because the subject is now nominative
• valet instead of valere
  because the verb is now finite, not infinitive

Then when Latin reports that statement after dicit, it changes to:

• honestatem ... valere

This direct-to-indirect shift is one of the most important patterns in Latin.

They are:

• iudex, iudicis = judge
• dico, dicere, dixi, dictum = say
• honestas, honestatis = honesty
• valeo, valere, valui = be strong, be well, be worth
• mendacium, mendacii = lie, falsehood

Knowing those dictionary forms helps explain the endings:

• iudex = nominative singular
• honestatem = accusative singular of honestas
• mendacium = nominative/accusative singular neuter

The big takeaway is this:

After a verb like dicit, Latin often reports a statement with accusative + infinitive.

So learn to recognize this pattern:

• main verb of saying/thinking
• then accusative subject
• then infinitive verb

In this sentence:

• Iudex dicit = main clause
• honestatem ... valere = indirect statement

If you can spot that pattern quickly, sentences like this become much easier to read.