Why island and not light?

In Northern Canon there is an Agama that talks about the lamp:

It was a dark night, raining lightly, with flashes of lightning. The Buddha said to Ananda: "You can come out with the umbrella over the lamp." Ananda listened, and walked behind the Buddha, with an umbrella over the lamp. When they reached a place, the Buddha smiled. Ananda said: "The Buddha doesn’t smile without a reason. What brings the smile today?" The Buddha said: "That’s right! That’s right! The Buddha doesn’t smile without a reason. Now you are following me with an umbrella over a lamp. I look around, and see everyone doing the same thing."

The idea is that each of us already have our own lamp, our basic sanity aka Buddha Nature. And yet, we all follow the Buddha and want him to show us the way.

Instead, the Buddha says, be your own lamp, use the light which you already have. As he says in Kalama Sutta - think, what actions bring what results. See for yourself.

In the dark, the flame of a single candle is an island of light, a refuge for those seeking light. Therefore to "be your own island" is to become like that island of light in the dark, a refuge:

Whether now or after I have passed, any who shall live as their own island, their own refuge, with no other refuge; with the teaching as their island and their refuge, with no other refuge—those mendicants of mine who want to train shall be among the best of the best.” SN47.13

The island analogy is from a dhammapada verse, wherein the island is not overcome by any flood. Sn45.171; At Savatthi.

"Monks, there are these four floods. Which four? The flood of sensuality, the flood of becoming, the flood of views, & the flood of ignorance. These are the four floods.

The path is for crossing over, is also in the 45.171 if i remember correctly. As to how to cross;

Snp1.1;
But how is it, dear sir, that by not halting and by not straining you crossed the flood?”

“When I came to a standstill, friend, then I sank; but when I struggled, then I got swept away. It is in this way, friend, that by not halting and by not straining I crossed the flood.”

That is the imagery. The light imagery comes into play as well, also in the Dhp ie; 'surrounded by darkness; shouldn't you search for light?'

In the Darkness Sutta sn56.46;

"There is, monks, an inter-cosmic[1] void, an unrestrained darkness, a pitch-black darkness, where even the light of the sun & moon — so mighty, so powerful — doesn't reach."

When this was said, one of the monks said to the Blessed One, "Wow, what a great darkness! What a really great darkness! Is there any darkness greater & more frightening than that?"

"There is, monk, a darkness greater & more frightening than that."

"And which darkness, lord, is greater & more frightening than that?"

"Any brahmans or contemplatives who do not know, as it actually is present, that 'This is stress'; who do not know, as it actually is present, that 'This is the origination of stress'... 'This is the cessation of stress'... 'This is the path of practice leading to the cessation of stress': They revel in (thought-) fabrications leading to birth; they revel in fabrications leading to aging; they revel in fabrications leading to death; they revel in fabrications leading to sorrow, lamentation, pain, distress, & despair. Reveling in fabrications leading to birth... aging... death... sorrow, lamentation, pain, distress, & despair, they fabricate fabrications leading to birth... aging... death... sorrow, lamentation, pain, distress, & despair. Fabricating fabrications leading to birth... aging... death... sorrow, lamentation, pain, distress, & despair, they drop into the darkness of birth. They drop into the darkness of aging... the darkness of death... darkness of sorrow, lamentation, pain, distress, & despair. They are not totally released from birth, aging, death, sorrows, lamentations, pains, distresses, & despairs. They are not totally released, I tell you, from suffering & stress.

So it can be seen that if ignorance is darkness then knowledge is light. This is often seen in the Sutta where people praise a good discourse for lighting a lamp for those with eyes to see.

Else such as Tamonata Sutta speaks of darkness and light as favorable and unfavorable conditions. Another example is snp2.6;

"A quarrelsome bhikkhu shrouded by delusion, does not comprehend the Dhamma taught by the Awakened One when it is revealed. Annoying those practiced in meditation, being led by ignorance, he is not aware that his defiled path leads to Niraya-hell. Falling headlong, passing from womb to womb, from darkness to (greater) darkness, such a bhikkhu undergoes suffering hereafter for certain.

Why does the sutta say "be your own island" instead of "be your own light"?

What is the significance of "be your own island"?

If you are on an island there is no other help. Likewise one or the dharma is one's own refuge.

The key here is the lack of other help.

You can have many lights illuminating, hence does not resonate that this is the only rescue.