Any test given would be compared with the results of previous tests before the mission or during training - this would give a picture of changes through time.
A good many tests would be physical, and their performance and attitudes during the physical tests would be telling of much. Notably the Chinese space agency's tests are particularly gruelling physically and in terms of mental endurance, but I'm having trouble digging up the details.
You've not defined what changes have happened, so I'll just give you a few tests to be going on with:
16 factor personality test (Raymond Cattell). (By interview).
Schizothymia (introversion, reduced affect, withdrawal from social contact)/affectothymia (easygoingness) axis.
General mental capacity, abstract thinking, concrete thinking, scholastic performance.
Emotional stability (with stimulus), calmness.
Dominance/submissiveness axis.
Livliness: Spontaneous, expressive, cheerful, happy-go-lucky/serious, introspective, silent, prudent.
Non-conforming/rule bound (super-ego strength).
Threctia/parmia (threat-sensitivity, high/low).
Harria: Utilitarian, objective, unsentimental, tough minded, self-reliant, no-nonsense, rough./Premsia: Sensitive, aesthetic, sentimental, tender minded, intuitive, refined.
Alaxia (trusting, unsuspecting)/premsia (suspicious, sceptical).
Praxernia (grounded, steady, practical)/autia (abstract, absent-minded, idea-absorbed).
Artlessness/shrewdness axis.
Untroubled/proneness to guilt.
Conservatism/radicalism.
Group-adherence/self-sufficiency.
Self-concept integration: Low (careless of social rules/norms of behaviour, impulsive, not well controlled)/high (conforming to group ways, controlled).
Ergic tension: Low (placid, relaxed, chilled-out)/ high (impatient, frustrated, tense, overwrought).
Likert scale: (survey of attitudes/preferences). Used regards to social/political attitudes, food preference, attitudes to other crew etc.. (Interview or written questionnaire).
Beck: depression inventory/anxiety inventory.
TAT, inkblot test: (measures feedback from external influences to determine unresolved emotional conflicts).
The good old Glasgow Coma Scale: measures response to stimuli - verbal, eye movement etc..
Direct observation: in a variety of settings.
Things to look for: delusions (false and fixed beliefs), obsessions, hallucinations (various sensory modalities), depression, alertness, concentration, abstract reasoning changes, signs of intoxication (disinhibition, impulse-control issues)..