The “i” doesn’t have a meaning by itself here, it is just part of the underlying structure - “ar ôl i <…>” = “after <…>”
Another construction with the same “i” would be “cyn i” = “before”, followed by an unconjugated clause. (For example: Pa mor hir wyt ti’n moyn mynd cyn i ni stopio am hoe? How long do you want to go before we stop for a rest?)
The “i Lerpwl” seems to be a typo to me, too. As such, I’d translate that sentence
One hundred and fifty years after the ship Mimosa sailed from Liverpool to Patagonia, Mari Gonzales came to Wales to see where her family used to live.
In addition to what @Hendrik said, there is a general “i” sentence construct in Welsh, along the lines of: brawddeg+i rhywun wneud rhybeth, for example:
“Mae Steffan yn awyddus i Hendrik fynd” - Steffan is keen for Hendrik to go,
“Meddyliodd Aled i Mair fynd adre” - Aled thought that Mair had gone home
This “i” construct is quite rich and useful, but too complex to do it justice here. “The Syntax of Welsh” describes it well. Our @garethrking says this about it: “it is not as alien to the spoken language as is sometimes claimed” (p.314, Modern Welsh: A Comprehensive Grammar"
Yes - these are two different constructions, actually. In the first, the i is simply the preposition that is governed by awyddus, while in the second we have the one that that idiot @garethrking correctly (for once) asserted to be OK in the spoken language - i.e. the one where i is used for (usually, though not exclusively) past-tense reported speech.
Glywes i i’r trên fynd hanner awr yn ôl I heard that the train had gone half an hour ago
(= Glywes i fod y trên wedi mynd hanner awr yn ôl)
Honnir mewn rhai o’n papurau newydd bore 'ma i’r llywodraeth fod ar fin syrthio It is claimed in some of our newspapers this morning that the government is about to fall
(= Honnir mewn rhai o’n papurau newydd bore 'ma fod y llywodraeth ar fin syrthio)