かけた as Current State
This question is testing whether you understand how the seemingly past tense かけた can actually be describing the current state of a person. This kind of verb usage happens a lot with articles of clothing.
The correct translation is not “who wore glasses”, but rather:
The person beside me who is wearing glasses is Suzuki-san.
めがねをかけている人 or めがねをかけた人 most often means the person who has glasses on, and not the person who is in the process of putting on glasses, nor the person who has put on glasses at some point. Similarly, 青いコートを着た人 usually just means person in a blue coat, not person who wore a blue coat.
かけた as Past Action
In contrast, the かけた in でんわをかけた人 clearly has a past tense meaning, i.e. person who made a phonecall.
As you point out, the でんわ choice is not ungrammatical per se (even though the context may be harder to imagine). However, the big hint is choice #2, かぎ. This noun acts similarly to でんわ, in that かぎをかけた人 means person who locked (a door). Since you cannot have multiple correct answers, these choices can be ruled out. Sadly, multiple choice questions often involve this type of sleuthing and mind-reading.
Why must で and かけている be used?
The textbook probably wanted to say that, in order to describe the person's current state and not an action done by them in the past, it must be 電話をかけている人 (person who is making a phonecall). Changing the となりの to となりで is not necessary, although the meaning will be altered a bit:
- 私のとなりのでんわをかけている人 [person beside me] who is making a call
- 私のとなりででんわをかけている人 person who is [making a call beside me]