The word "that" in the sentence "The time of the day that we take medicine makes a difference to how successful the treatment will be" functions as a relative pronoun introducing a restrictive (defining) relative clause: "that we take medicine."

1. Grammatical Function of "that"
Type: Relative Pronoun (not a conjunction)
Function: It introduces a restrictive relative clause that provides essential information about "the time of the day."
Antecedent: "The time of the day"
Meaning: It specifies "which time of the day" is being discussed.

2. Sentence Breakdown
Main Clause: The time of the day ... makes a difference to how successful the treatment will be.
Relative Clause: that we take medicine (modifies "the time of the day").
Verb in Relative Clause: "take"

3. Comparison with "The time of the day when we take medicine"
An alternative way to express this sentence is:  "The time of the day when we take medicine makes a difference to how successful the treatment will be."
Here, "when" is an adverbial relative pronoun indicating time, replacing "that."

4.Summary
"That" is a relative pronoun, not a conjunction or an adverb.
It introduces an essential relative clause ("that we take medicine") modifying "the time of the day."
"When" can be used instead of "that" for clarity.
In formal writing, "when" is typically preferred over "that" when referring to time.