Pathfinder has many options for giving a player an weapon that becomes stronger as they do or ways to grant that flavor.
I rank these options starting with the one that takes the least amount of work for you.
1. Allow players to upgrade their equipment
You could allow players to upgrade equipment by having them pay the difference in price -- so upgrading a +1 weapon to a +2 weapon requires paying 6,000 gp. This eschews all the extra annoyance of having item creation feats or paying an NPC to do it.
This is such a common house rule that even the official organized play program (Pathfinder Society) uses this method.
Your player could reflavor this process by having the PC spend gold on reagents for a ceremony that properly attunes the weapon to his current power.
2. Use an archetype with a scaling weapon
A number of archetypes and player options grant a PC a weapon that scales with level. Examples of these include:
- Bladebound Magus, a melee combat spellcaster that gains a sentient weapon that grows in power
- Blade Adept, an arcanist that has the bladebound magus's weapon
- Jistkan Artificer, a melee combat spellcaster with a golem arm that gets stronger with levels
- Gloomblade, a fighter that has a shapeshifting shadow weapon that gets more powerful with levels
- Phantom Blade, an occult spellcaster with a ghostly weapon
A number of 3rd party options like Dreamscarred Press's soul knife would also fit the flavor.
3. Grant a scaling magic weapon
Pathfinder Unchained introduces the Scaling Item optional rules. These rules allow a PC to have a magic item that increases in price and power as the character level increases. This comes at the cost of less wealth for the PC, so you would have to give this PC less gold.
You can use one of the sample weapons from the book or you could create the weapon yourself with the below recommendations.
According to Table 4-13: Value of Scaling Items, a magic weapon would be a "wonder" that accounts for 30% of the character's wealth. With this, we can estimate the weapon will scale accordingly:
- At 2nd level, the item becomes a masterwork weapon
- At 4th level, the item becomes a +1 weapon
- Between 7th and 8th level, the item becomes a +2 weapon
- At 10th level, the item becomes a +3 weapon
- At 12th level, the item becomes a +4 weapon
- At 14th level, the item becomes a +5 weapon
- At 15th level, the item costs as much as a +6 weapon
- At 16th level, the item costs as much as a +7 weapon
- At 17th level, the item costs as much as a +8 weapon
- At 18th level, the item costs as much as a +9 weapon
- At 19th level, the item costs as much as a +10 weapon
Note that a weapon's enhancement bonus generally cannot go higher than a +5. So it would need additional abilities if the weapon's total price is greater than that of a +5 weapon.
4. Use Automatic Bonus Progression
This is the "nuclear" option. Pathfinder Unchained's Automatic Bonus Progression eschews bonus-granting magic items entirely in favor of automatically granting the bonuses to the PCs. This comes at the cost of reducing wealth by level.
Though meant to apply to the entire party, you could possibly apply the weapon attunement ability to the specific PC at the cost of about 25% of his wealth.
The character can attune herself to any one weapon in her possession, and can change that attunement once per day. The attuned weapon gains a +1 enhancement bonus at 4th level. At 8th level, the character can split her attunement between two weapons, granting each a +1 enhancement bonus. At 9th level, she can grant a single weapon a +2 enhancement bonus instead of granting two weapons a +1 enhancement bonus each. At 14th level, she can either grant a single weapon a +3 enhancement bonus or grant two weapons a +2 enhancement bonus each. At 15th level, she can either grant a single weapon a +4 enhancement bonus or grant two weapons a +3 enhancement bonus each. At 17th level, she can either grant a single weapon a +5 enhancement bonus or grant one weapon a +4 enhancement bonus and another weapon a +3 enhancement bonus.
If you use Automatic Bonus Progression, I recommend treating the PCs as 1 level higher (so a 2nd level PC gets the 3rd level benefits). As someone who uses this optional rule in his home game, I've found the basic progression to be too slow and have PCs end up with bonuses too low for their level. Level + 1 feels about right.