1)「ボクにもできる。」
2) "I can also do that."
While those two sentences may be good "translations" for each other, they are structured very differently. It is, indeed, "translation" that often gets in the way of understanding things between the two languages.
The English sentence clearly has a grammatical subject in "I", but the subject is unmentioned in the Japanese sentence. 「ボク」 is not the subject of sentence #1, believe it or not.
「ボクにもできる。」 literally means "~~ is doable for me, too."
The ~~ part is just not mentioned because it is understood between the speaker and listener from the larger context/situation. If one is to actually mention and use it in the sentence, one would say:
「ボクにも~~ができる。」
Notice that 「ボク」 is still not the subject of that sentence; 「~~」 is. It is saying "~~ is doable for me, too.", but only for the sake of naturalness of the translated English sentence, one might use "I can also do ~~", "I can do ~~, too", etc.
Long preface, I know, but from my experience with Japanese-learners, I just know how common this misunderstanding is among them.
Thus, 「に」 is like "to" or "for" as in "X is Y to/for me" in English. Once again, this 「に」 does not mark the subject.
This 「に」 is the signature particle used in phrases such as:
「Person + に + わかる/できる/[見]{み}える/[言]{い}える, etc.」
Finally, the 「も」 is there for emphasis. It means "even" here.
In essence, the original sentence literally means:
"~~ is doable even for me."