Fill in the blank with the most suitable word from the options given…

The correct answer is Option (1) → Alternate

Adjective before a noun:

• The blank stands before the noun "suggestions", so it needs an adjective that describes them.
• Alternate, used as an adjective, here means offering another choice (the same as "alternative", a use common in American English).
• This sentence: the Principal gave the students other ways to settle the matter and asked them to call off the strike.
• Complete sentence: The Principal gave some alternate suggestions to the students and asked them to call off the strike.
  Example: The road was closed, so we took an alternate route.

Why the Other Options are Incorrect?

Option 2: Alternating — Incorrect. Alternating means changing by turns, first one and then the other; "alternating suggestions" does not describe advice offered as another choice.
Example: The lights flashed in alternating colours.

Option 3: Alternately — Incorrect. Alternately is an adverb (by turns); an adverb cannot describe the noun "suggestions".
Example: The two teams batted alternately.

Option 4: Altering — Incorrect. Altering is the -ing form of alter (to change something); "altering suggestions" makes no sense here.
Example: The tailor is altering the dress.