Is the complement considered a part of the predicate?

A predicate, defined, is the part of the sentence that modifies the subject. For example:

He ran quickly. (ran quickly is the predicate)
Ben read the book (read the book modifies the subject, and is hence, the predicate)

Having defined a predicate, let's analyse your sentence.
"the strangest person" describes the subject, "he". Thus, it is the predicate.
What about "she had ever met"? "She had ever met" is a dependent clause, that is modifying "strangest person". As it is not directly modifying the subject, but modifying something that modifies the subject, it's called the "secondary predicate":

A secondary predicate is a (mostly adjectival) predicative expression that conveys information about the subject but is not the main predicate of the clause. This structure may be analysed in many different ways.
These may be resultative, as in (1) and (2) or descriptive as in (3).
(1) She painted the town red
(2) The film left me cold

As "the strangest person she had ever met" was the complement, as well as the predicate, the complement in this case is part of the predicate.

Assuming complement means an argument of the verb other than the subject, and if you analyze sentences into subject and predicate, then a complement must be part of the predicate. There's nowhere else for it to be.

For instance, if you split a sentence like "John loves Mary" into two parts, subject "John" and predicate "loves Mary", as is customary in traditional grammar, the complement to the verb, "Mary", has to be part of the predicate.

In your example, "He was the strangest person she had ever met", since "he" is the subject, the predicate must be "was the strangest person she had ever met". (I'm don't know how the term "link word" is used). That makes the complement to "was", "the strangest person she had ever met", a part of the predicate.

In predicate logic, the term "predicate" is used differently. You could still call "loves Mary" in the above example a predicate, but predicates can have several arguments, so "loves" counts as a predicate with two arguments, the subject "John" and the complement object "Mary". Using terms this way, a complement is not part of the predicate.

For your example, the "was" probably would not be treated as a predicate by a logician. There is a variety of predicate logic with a special identity operation, and maybe that could be used to represent "was", though the past tense would be a problem. I'm not sure that your example even has a predicate in the sense in which that term is used in predicate logic.