lord of the rings - When and why did Bilbo write a song about Aragorn

There's absolutely no indication in the books that Bilbo and Aragorn met in 2941. This would have been during Bilbo's stop in Rivendell on the Quest of Erebor, when Aragorn also lived there as a child, so it's a fan surmise that they may have met, but not one based on anything Tolkien actually wrote.

Also note Bilbo's words about the poem (Council of Elrond, with my emphasis):

'I made that up myself,' he whispered to Frodo, 'for the Dúnadan, a long time ago when he first told me about himself. I almost wish that my adventures were not over, and that I could go with him when his day comes.'

So Bilbo didn't write the poem when they first met, but rather when Aragorn first told Bilbo about himself. Now, Bilbo and the Dwarves were in Rivendell in 2941, but the Tale of Years records that Aragorn was not told his true identity and lineage until 2951 (10 years later):

Elrond reveals to 'Estel' his true name and ancestry, and delivers to him the shards of Narsil.

So even if Bilbo had met Aragorn in 2941, he could not have written the poem then, because Aragorn could not have told Bilbo about himself then, because Aragorn didn't even know about himself then. The answer to "why would anyone write a song for a 10 year old kid" is therefore "they didn't".

The actual date of their first meeting is unrecorded, but we can narrow down the range of possibilities and arrive at a probable date.

The Tale of Years records that Bilbo, following his long-expected party, settled in Rivendell in 3002 but that from 3009 onwards Aragorn was engaged (with Gandalf) in the search for Gollum, who was finally captured (by Aragorn) in 3017. So the most likely date for their first meeting is sometime in that range - 3002 to 3009 - but Aragorn's movements during that time are unrecorded, aside for 3006/3007 when Aragorn was known to be in the north and met his mother before her death.

So Bilbo almost certainly would not have had the opportunity to meet Aragorn before 3002, and the strongest possibility for the year of their first meeting seems to be 3006/3007.

When Bilbo says "a long time ago" to Frodo, he's actually talking about (most likely) no more than 16 years previously, with a strong possibility that it was only 11/12 years previously.